What is your favorite "strange" function? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-18T21:11:09Z http://mathoverflow.net/feeds/question/22189 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function What is your favorite "strange" function? vonjd 2010-04-22T14:16:37Z 2012-07-17T15:40:43Z <p>There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you consider strange and especially like.</p> <p>Please also give a reason why you find this function strange and why you like it. Perhaps you could also give some kind of reference where to find further information.</p> <p>As usually: Please only mention one function per post - and let the votes decide :-)</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22191#22191 Answer by Nate Eldredge for What is your favorite "strange" function? Nate Eldredge 2010-04-22T14:45:34Z 2010-04-22T14:45:34Z <p><strong>An unbounded operator with dense graph.</strong></p> <p>In functional analysis, one deals with unbounded operators on Hilbert space, but usually ones that are closed, or are at least closable. At the opposite end of the spectrum, one can construct linear operators whose graph is dense: for any pair $(x,y)$, there is a sequence $x_n$ such that $x_n \to x$ and $A x_n \to y$ ! It's not so easy even on $\mathbb{R}$ to come up with a function whose graph is dense, and the examples I think of aren't measurable. But in infinite dimensions, you can find one that is linear! It's just an illustration that facts that are trivial in finite dimensions can be horribly, horribly false in infinite dimensions.</p> <p>A family of examples of this is constructed in</p> <p>MR0782615 (86i:47052) Lindsay, J. M. A family of operators with everywhere dense graphs. Exposition. Math. 2 (1984), no. 4, 375--378. </p> <p>Interestingly, as an appplication, Lindsay uses such operators to prove that a <strong>Brownian motion sample path</strong> is nowhere differentiable --- which is my <em>other</em> favorite strange function!</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22192#22192 Answer by J.C. Ottem for What is your favorite "strange" function? J.C. Ottem 2010-04-22T14:46:35Z 2010-04-22T14:46:35Z <p>I like the <a href="http://en.wikipedia.org/wiki/Theta_function" rel="nofollow">Theta functions</a> which are given by Fourier-type series. They show up in many areas in mathematics. For example:</p> <p>i)They are very important in the study of abelian varieties in algebraic geometry (for example, in the case of elliptic curves they are used in the proof of <a href="http://en.wikipedia.org/wiki/Abel%25E2%2580%2593Jacobi_map" rel="nofollow">Abel's theorem</a> and are related to Weierstrass $\mathcal{P}-$function). </p> <p>ii) They satisfy a number of interesting indentities. For example, in the one-dimensional case, they satisfy Jacobi's triple product identity which can be used to show <a href="http://en.wikipedia.org/wiki/Jacobi%2527s_four-square_theorem" rel="nofollow">Jacobi's four square theorem</a></p> <p>iii) They can be used to solve algebraic equations degree equations explicitly (see <a href="http://library.wolfram.com/examples/quintic/main.html" rel="nofollow">this link</a>)</p> <p>iv) In the one-dimensional case, they solve the heat equation. </p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22194#22194 Answer by Sunni for What is your favorite "strange" function? Sunni 2010-04-22T14:52:15Z 2010-04-22T14:52:15Z <p>Dirac delta function seems strange to me since the first time I saw it. <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" rel="nofollow">http://en.wikipedia.org/wiki/Dirac_delta_function</a></p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22200#22200 Answer by Gabriel Benamy for What is your favorite "strange" function? Gabriel Benamy 2010-04-22T15:32:06Z 2010-04-22T15:32:06Z <p>The <a href="http://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow">Weierstrass function</a> is particularly intriguing, as it's a function that's everywhere continuous, but nowhere differentiable.<br> $f(x)= \sum_{n=0} ^\infty a^n \cos(b^n \pi x)$<br> where 0&lt;<em>a</em>&lt;1, and <em>b</em> is a positive odd integer such that $ab > 1 + \frac{3\pi}{2}$.<br> It challenges the notion that, just because a function is continuous, it must also be differentiable in most places, which I think is pretty cool.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22201#22201 Answer by Christian Blatter for What is your favorite "strange" function? Christian Blatter 2010-04-22T15:35:15Z 2010-04-22T15:35:15Z <p>Among the "special" functions encountered in analysis in my view Ingrid Daubechies' waveletes with compact support are the strangest.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22205#22205 Answer by Dinesh for What is your favorite "strange" function? Dinesh 2010-04-22T15:47:14Z 2010-04-22T15:47:14Z <p>Minkowski's question mark function if only for the strange $?(\cdot)$ notation.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22206#22206 Answer by Nate Eldredge for What is your favorite "strange" function? Nate Eldredge 2010-04-22T15:51:11Z 2010-11-13T15:01:37Z <p>A <strong><a href="http://en.wikipedia.org/wiki/Wiener_process#Some_properties_of_sample_paths" rel="nofollow">Brownian motion sample path</a></strong>.</p> <p>These are about the most bizarrely behaved continuous functions on $\mathbb{R}^+$ that you can think of. They are nowhere differentiable, have unbounded variation, attain local maxima and minima in every interval... Many, many papers and books have been written about their strange properties.</p> <p>Edit: As commented, I should clarify the term "sample path". Brownian motion is a stochastic process $B_t$. We say a sample path of Brownian motion has some property if the function $t \mapsto B_t$ has that property almost surely. So, run a Brownian motion, and with probability 1 you will get a function with all these weird properties.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22218#22218 Answer by Antonio E. Porreca for What is your favorite "strange" function? Antonio E. Porreca 2010-04-22T16:37:49Z 2010-04-22T16:37:49Z <p><strong>The Busy Beaver function</strong></p> <p>Let &Sigma; be a finite alphabet, for instance {0, 1}; let <b>M</b> be the set of Turing machines with alphabet &Sigma;, and let <b>H</b> &sube; <b>M</b> be the set of Turing machines that halt when given the empty string &epsilon; as input.</p> <p>For each <i>M</i> &isin; <b>H</b>, Let <i>s</i>(<i>M</i>) be the number of steps performed by <i>M</i> before halting (when given &epsilon; as input).</p> <p>Finally, let <i>S</i> : &#8469; &rarr; &#8469; be the function defined by</p> <blockquote> <p><i>S</i>(<i>n</i>) = max {<i>s</i>(<i>M</i>) : <i>M</i> &isin; <b>H</b> and <i>M</i> has <i>n</i> states}</p> </blockquote> <p>Notice that <i>S</i> is well-defined, since only finitely many Turing machines with <i>n</i> states exist.</p> <p>In other words, <i>S</i>(<i>n</i>) is the maximum number of steps performed on &epsilon; among all halting Turing machines with <i>n</i> states. <i>S</i> is called the <a href="http://en.wikipedia.org/wiki/Busy_beaver" rel="nofollow">Busy Beaver function</a>.</p> <p>It turns out that <i>S</i> is uncomputable because it grows faster than any computable function, that is, for all recursive functions <i>f</i> : &#8469; &rarr; &#8469; we have <i>S</i>(<i>n</i>) > <i>f</i>(<i>n</i>) for large enough <i>n</i>, and in particular <i>f</i> is <i>o</i>(<i>S</i>).</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22223#22223 Answer by Glen M Wilson for What is your favorite "strange" function? Glen M Wilson 2010-04-22T17:19:14Z 2010-04-22T17:19:14Z <p>I like the Cantor function. A continuous, increasing function $f:[0,1]\rightarrow[0,1]$ with derivative $0$ almost everywhere. See wiki article <a href="http://en.wikipedia.org/wiki/Cantor_function" rel="nofollow">here</a>. </p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22229#22229 Answer by Julián Aguirre for What is your favorite "strange" function? Julián Aguirre 2010-04-22T18:06:17Z 2010-04-22T18:06:17Z <p>Please excuse me if I include two related functions in one answer. Any space filling curve is rather strange, at least for me. Let $\gamma\colon[0,1]\to[0,1]^2$ be such a curve, that is, $\gamma$ is continuous and surjective. Let $\gamma(t)=(x(t),y(t))$; then $x(t)$ (or $y(t)$) is my other candidate for strangest function: given any $z\in[0,1]$, $x^{-1}(z)$ has the cardinality of the continuum.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22231#22231 Answer by Jacques Carette for What is your favorite "strange" function? Jacques Carette 2010-04-22T18:21:44Z 2010-04-22T18:21:44Z <p>One can construct a natural 'metric' for the Riemann sphere which is equivalent to the spherical metric but which is singular on a dense set of points of the Riemann sphere though remains $L^1$ integrable.</p> <p>These are built from degree 2 rational maps (first constructed by Mary Rees) which have the whole Riemann sphere as their Julia sets, and have the orbits of their critical points also dense. The Carlesson-Jones-Yoccoz construction of a expanding metric for critically-finite rational maps actually extends to this case, and we get a metric in which this Julia set actually <em>looks</em> as if it was hyperbolic!</p> <p>[The details are worked out in my PhD thesis, never published as I decided that computer algebra suited me better than complex dynamics].</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22236#22236 Answer by Tom Smith for What is your favorite "strange" function? Tom Smith 2010-04-22T18:57:09Z 2010-04-22T18:57:09Z <p>The empty function $\emptyset:\emptyset\to\emptyset$ is quite strange when you first meet it.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22239#22239 Answer by Joel David Hamkins for What is your favorite "strange" function? Joel David Hamkins 2010-04-22T19:14:08Z 2010-04-22T19:14:08Z <p>The <a href="http://en.wikipedia.org/wiki/Ackermann_function" rel="nofollow">Ackermann function</a> $A(n,m)$ is defined on the natural numbers by a very simple recursion, but the values grow enormously, almost beyond conception. This function completely transcends any simple-minded system of rates-of-growth based on polynomial, exponential, double-exponential and so on. </p> <p>The first few values of the diagonal function $A(n) = A(n,n)$ are:</p> <ul> <li>$A(0) = 1$</li> <li>$A(1) = 3$</li> <li>$A(2) = 7$</li> <li>$A(3) = 61$</li> <li>$A(4) = 2^{2^{2^{65536}}}-3$</li> <li>$A(5)$ is vast, and can be described in terms of exponential stacks of $2$s, whose height is a stack of $2$s, etc. 5 times. </li> <li>$A(6)$ is so vast, it is best described using the Ackermann function itself.</li> </ul> <p>The levels of the Ackerman function $A_n(m)=A(n,m)$ stratify the primitive recursive functions, in the sense that they are each primitive recursive, but every primitive recursive function is bounded by such a level of the Ackermann function. Thus, the Ackermann function itself is not primitive recursive, although it is computable in the sense of computability theory.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22259#22259 Answer by kakaz for What is your favorite "strange" function? kakaz 2010-04-22T21:10:15Z 2010-04-22T21:10:15Z <p>I suppose the strangest function in mainstream mathematics is <em>Riemann zeta function</em> <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">http://en.wikipedia.org/wiki/Riemann_zeta_function</a></p> <p>$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \;\;\;\;\;\;\;{Re}(s) >1.$</p> <p>It is part of one of the most important hypothesis and is very influential in many branch of moder mathematics. It is actively used in many areas and is researched in many ways, it is not curiosity, or exotic example, but important mathematical being! </p> <p>And is mysterious and strange! Take a look: <img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Complex_zeta.jpg/600px-Complex_zeta.jpg" alt="alt text"></p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22305#22305 Answer by Jon for What is your favorite "strange" function? Jon 2010-04-23T06:09:30Z 2010-07-01T18:31:14Z <p>It is pretty obvious after you've seen it, but I like the crinkled curve from Halmos's Hilbert Space Problem book:</p> <p>Let $f:\mathbb{R}\rightarrow(0,\infty)$ be an $L^2$ function, and define $t\mapsto g_t:\mathbb{R}\rightarrow L^2(\mathbb{R})$ by $$g_t(x)=\chi_{(-\infty,t)}(x) \times f(x).$$</p> <p>Then $g_t$ has the property that for all $t_1 &lt; t_2 &lt; t_3$ the secants $g_{t_2}-g_{t_1}$ and $g_{t_3}-g_{t_2}$ are mutually orthogonal. (The curve turns a corner at every point.)</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22307#22307 Answer by Mariano Suárez-Alvarez for What is your favorite "strange" function? Mariano Suárez-Alvarez 2010-04-23T06:23:47Z 2010-04-23T06:23:47Z <p>I'm still quite impressed about $f(x)=\mathrm e^x$ &hellip;</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22308#22308 Answer by Jon for What is your favorite "strange" function? Jon 2010-04-23T06:56:18Z 2010-11-13T14:51:14Z <p>The Osgood curve ("<a href="http://www.jstor.org/stable/1986455" rel="nofollow">A Jordan Curve of Positive Area</a>") is an injective map from [0,1] into $\mathbb{R}^2$ which traces out an image of positive area. (This differs from standard space-filling curves, which are not injective.)</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22318#22318 Answer by Yaakov Baruch for What is your favorite "strange" function? Yaakov Baruch 2010-04-23T08:38:40Z 2010-04-23T08:38:40Z <p>Since Mariano took my favorite already, I'll go with the stopping time function for the 3x+1 problem: <a href="http://www.ieeta.pt/~tos/3x+1.html" rel="nofollow">http://www.ieeta.pt/~tos/3x+1.html</a></p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22340#22340 Answer by Keivan Karai for What is your favorite "strange" function? Keivan Karai 2010-04-23T12:32:01Z 2010-04-23T12:32:01Z <p>Interpreting your questions a bit liberally, I suggest the Goodstein sequence:</p> <p><a href="http://en.wikipedia.org/wiki/Goodstein%27s_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Goodstein%27s_theorem</a></p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22354#22354 Answer by Joseph Malkevitch for What is your favorite "strange" function? Joseph Malkevitch 2010-04-23T14:18:47Z 2010-04-23T14:18:47Z <p>f(x) = sin (1/x): (x not 0); f(x) = 0 (x equals 0)</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/30245#30245 Answer by Colin Pratt for What is your favorite "strange" function? Colin Pratt 2010-07-01T22:57:59Z 2010-11-13T15:48:50Z <p>A canonical example from elementary real analysis - the Blancmange function. Consider $f$ defined piecewise by</p> <p>$f(x) = x - [x], \quad \text{if} \quad 0 \leq x- [x] \leq \frac{1}{2}$,</p> <p>and</p> <p>$f(x) = 1 - (x - [x]), \quad \text{if} \quad \frac{1}{2} &lt; x - [x] &lt; 1$,</p> <p>(where $[x]$ is the integer part of $x$). Then define the Blancmange function, $B$ </p> <p>$B(x) = \sum_{n=0}^{\infty}\dfrac{1}{2^n} f(2^{n}x)$.</p> <p>The series converges by the Comparison Test, since $|f(2^{n}x)| \leq \frac{1}{2}$, for all $x \in \mathbb{R}$, and it can be shown that $B$ is <em>uniformly continuous</em> but <em>nowhere differentiable</em>. Here a picture of the function:</p> <p><img src="http://upload.wikimedia.org/wikipedia/commons/5/54/Blancmange-function.svg" alt="Blancmange function"></p> <p>A tasty counterexample to the converse of "differentiability $\implies$ continuity".</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/30252#30252 Answer by Gerry Myerson for What is your favorite "strange" function? Gerry Myerson 2010-07-02T01:07:15Z 2010-07-02T01:07:15Z <p>These functions like the Cantor function and the continuous-but-not-differentiable function are all well and good, but contrived - the only place you ever see them is as counterexamples. Here is a function that has many uses in Number Theory, and still manages to have a strange property or two. Let $x=h/k$ with $h$ and $k$ integers, $k>0$. Define $$s(x)=\sum_{c=1}^{k-1}((c/k))((ch/k))$$ where $((y))=0$ if $y$ is an integer, $((y))=\lbrace y\rbrace-1/2$ otherwise. It is easily proved that the sum depends only on the ratio of $h$ and $k$, not on their individual values, so $s$ is a well-defined function from the rationals to the rationals. It is known as the Dedekind sum; it came up originally in Dedekind's study of the transformation formula of the Dedekind $\eta$-function. </p> <p>Now for the strange properties. </p> <p>Hickerson, Continued fractions and density results for Dedekind sums, J Reine Angew Math 290 (1977) 113-116, MR 55 #12611, proved that the graph of $s$ is dense in the plane. </p> <p>With Nick Phillips, I proved (Lines full of Dedekind sums, Bull London Math Soc 36 (2004) 547-552, MR 2005m:11075) that, with the exception of the line $y=x/12$, every line through the origin with rational slope passes through infinitely many points on the graph of $s$. We suspect that the points are dense on those lines, though we could only prove it for the line $y=x$. </p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/30253#30253 Answer by Carl Mummert for What is your favorite "strange" function? Carl Mummert 2010-07-02T01:12:01Z 2010-07-02T01:12:01Z <p>The <strong>Banach limit</strong> assigns to every bounded sequence of real numbers a real number "limit" in a way that is linear, shift invariant, and agrees with the usual limit whenever it exists. Banach limits are among the mysterious examples of continuous linear functionals on $\ell^\infty$ that aren't represented by elements of $\ell^1$. Unfortunately, the Hahn-Banach theorem is used in the construction of the Banach limit, and the values aren't canonical. There's a precise definition at <a href="http://en.wikipedia.org/wiki/Banach_limit" rel="nofollow">http://en.wikipedia.org/wiki/Banach_limit</a> . </p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/30375#30375 Answer by Allen Knutson for What is your favorite "strange" function? Allen Knutson 2010-07-03T05:44:51Z 2010-07-03T05:44:51Z <p>Fix a probability $p &lt; 1/2$ of winning an unfair coin toss. For $x \in [0,1]$ rational, let $f(x)$ be the probability that, if you started with $x$ dollars, you could make it to 1 dollar through optimal betting* on the outcome of these coin flips. This function $f(x)$ is obviously weakly increasing on $[0,1]$ (in fact strictly). Less obvious is that it extends to a continuous function on $[0,1]$, whose derivative exists almost everywhere, but that derivative is $0$.</p> <p><a href="http://www.maa.org/joma/Volume8/Siegrist/RedBlack.pdf" rel="nofollow">http://www.maa.org/joma/Volume8/Siegrist/RedBlack.pdf</a></p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/36311#36311 Answer by Daniel Miller for What is your favorite "strange" function? Daniel Miller 2010-08-21T18:31:00Z 2010-08-21T18:31:00Z <p>Any of the isomorphisms $\mathbb{C}'\to S^{1}$, where $S^{1}$ is the unit circle and $\mathbb{C}'$ is the non-zero complex numbers, with the group operation for both being multiplication.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/36324#36324 Answer by paul Monsky for What is your favorite "strange" function? paul Monsky 2010-08-21T20:03:17Z 2010-08-21T20:03:17Z <p>Characteristic p commutative algebra leads naturally to the construction of various continuous functions on [0,1]^m that have beautiful self-similarity properties; for explication and some pictures see:</p> <pre><code>Pedro Teixeira, Syzygy gap fractals--I, arXiv 1008.0583 </code></pre> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/36325#36325 Answer by Thierry Zell for What is your favorite "strange" function? Thierry Zell 2010-08-21T20:16:44Z 2010-08-22T04:13:11Z <p>Just a simple construction to illustrate <a href="http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22191#22191" rel="nofollow">Nate Eldredge's answer</a> about functions with dense graphs. Pick any $\mathbb{R}$-vector space <em>E</em> with a norm. On <em>E</em>, choose a <em>non-continuous</em> linear form $L: E \to \mathbb{R}$; now this can only be done if $\dim(E)=\infty$, of course. </p> <p>Then, pick <em>y</em> such that $L(y)=1$, and let $T: E \to E$ be defined by $Tx=x-L(x)y$. Then obviously <em>T</em> maps <em>E</em> onto the kernel of <em>L</em>; it is not difficult to prove that $\ker (L)$ must be dense in <em>E</em> for any non-continuous <em>L</em> (the two conditions are even equivalent), and thus the graph of <em>T</em> must be dense in $E \times E$.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/36328#36328 Answer by Gerald Edgar for What is your favorite "strange" function? Gerald Edgar 2010-08-21T21:14:33Z 2010-08-21T21:14:33Z <p><a href="http://mathoverflow.net/questions/16829/what-are-your-favorite-instructional-counterexamples/17285#17285" rel="nofollow">The Fabius function</a></p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/45885#45885 Answer by Andrew Parker for What is your favorite "strange" function? Andrew Parker 2010-11-13T01:01:12Z 2010-11-13T01:01:12Z <p>I can't believe no one has mentioned the <A HREF="http://mathworld.wolfram.com/DirichletFunction.html" rel="nofollow">Dirichlet function</A>: <IMG SRC="http://mathworld.wolfram.com/images/eps-gif/DirichletFunction_1000.gif"><BR> (I guess it's up to me to bring it up...)</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/45887#45887 Answer by A Rannow for What is your favorite "strange" function? A Rannow 2010-11-13T01:45:24Z 2010-11-13T01:45:24Z <p>How about a function f: f(f(x)) = exp(x).</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/45889#45889 Answer by Peter Shor for What is your favorite "strange" function? Peter Shor 2010-11-13T01:57:07Z 2010-11-13T01:57:07Z <p>How about the function given by the Banach-Tarski paradox? This maps a ball into two copies of the same size ball, and is composed of isometries on subsets of $\mathbb{R}^3$.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/45890#45890 Answer by Tom Goodwillie for What is your favorite "strange" function? Tom Goodwillie 2010-11-13T02:24:03Z 2010-11-13T13:44:13Z <p>The function defined by the power series $f(x)=x-x^2+x^4-x^8+x^{16}-\dots$ What is its limit as $x$ approaches $1$ from below? EDIT (This answer is a trick question.)</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/45914#45914 Answer by Koundinya Vajjha for What is your favorite "strange" function? Koundinya Vajjha 2010-11-13T12:34:34Z 2010-11-13T15:51:19Z <p><a href="http://en.wikipedia.org/wiki/Thomae%27s_function" rel="nofollow">Thomae's function</a>, also called the "popcorn function". It's continuous at all irrationals and discontinuous at all rationals. Here a picture:</p> <p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/15/Thomae_function_%280,1%29.svg/200px-Thomae_function_%280,1%29.svg.png" alt="Thomae's function"></p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/45939#45939 Answer by Richard Stanley for What is your favorite "strange" function? Richard Stanley 2010-11-13T17:32:34Z 2010-11-13T17:32:34Z <p>Nonconstant continuous locally recurrent functions are quite unintuitive. A real-valued function is <em>locally recurrent</em> on $\mathbb R$ if for every $x_0\in\mathbb R$ and every deleted neighborhood $N(x_0)$ of $x_0$, there exists $x\in N(x_0)$ for which $f(x)=f(x_0)$. Thus in some sense a nonconstant continuous locally recurrent function looks everywhere like $x\sin(1/x)$ at $x=0$. See papers in the <em>American Math. Monthly</em> of Bush (1962), Marcus (1963), and Mauldon (1965). </p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/46161#46161 Answer by secretman for What is your favorite "strange" function? secretman 2010-11-15T22:56:07Z 2010-11-15T22:56:07Z <p>The <a href="http://en.wikipedia.org/wiki/Conway_base_13_function" rel="nofollow">Conway base 13 function</a> has to be the weirdest function I know. This function is continuous nowhere, yet it satisfies the intermediate value theorem. Only John Conway...</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/102367#102367 Answer by PaPiro for What is your favorite "strange" function? PaPiro 2012-07-16T16:45:13Z 2012-07-16T21:32:45Z <p>I like the beauty and <em>mysticism</em> of Euler's identity:</p> <p>$$f(\theta) = e^{i\theta} = \cos\theta + i \sin\theta$$</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/102460#102460 Answer by dab for What is your favorite "strange" function? dab 2012-07-17T15:40:43Z 2012-07-17T15:40:43Z <p>The formula for the nth term in the Fibonacci sequence</p> <p>$F_{n} = \cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1+\sqrt{5}}{2}\right)^n-\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1-\sqrt{5}}{2}\right)^n$</p> <p>This is interesting because it is a non-recursive expression for the Fibonacci sequence and also because it involves the golden ratio.</p>