Examples of badly behaved derivatives - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:26:14Z http://mathoverflow.net/feeds/question/34264 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34264/examples-of-badly-behaved-derivatives Examples of badly behaved derivatives acarchau 2010-08-02T14:44:22Z 2011-12-13T14:08:31Z <p>Consider a real valued function g on an open interval $(a,b)$ which is the derivative of a function continuous on $[a,b]$ at each point of $(a,b)$. The function $g$ has the intermediate value property, so a monotone $g$ will have to be continuous, a general $g$ cannot have simple discontinuities, etc. With such constraints how badly can a derivative behave in terms of continuity, can it get much worse than the derivative of say, $x^2 \sin(1/x)$?</p> http://mathoverflow.net/questions/34264/examples-of-badly-behaved-derivatives/34270#34270 Answer by Nate Eldredge for Examples of badly behaved derivatives Nate Eldredge 2010-08-02T15:26:02Z 2010-08-02T15:26:02Z <p>Derivatives are like continuous functions in that both have the intermediate value property, but they need not have the extreme value property. Gelbaum and Olmstead, <em>Counterexamples in Analysis</em>, gives as example 3.7 the function <code>$$f(x) = \begin{cases} x^4 e^{-\frac{1}{4}x^2}\sin\frac{8}{x^3}, &amp; x \ne 0 \\ 0, &amp; x=0 \end{cases}$$</code> for which $f'$ exists everywhere, but does not achieve its supremum on the compact interval $[-1,1]$.</p> <p>On the other hand, a derivative cannot have too many discontinuities. Lars Olsen at <a href="http://mathforum.org/kb/message.jspa?messageID=281579&amp;tstart=0" rel="nofollow">http://mathforum.org/kb/message.jspa?messageID=281579&amp;tstart=0</a> points out that a derivative is a pointwise limit of the continuous functions and hence of Baire class at most 1, and therefore its set of discontinuities is meager (in particular, has dense complement).</p> http://mathoverflow.net/questions/34264/examples-of-badly-behaved-derivatives/34273#34273 Answer by Ilmari Karonen for Examples of badly behaved derivatives Ilmari Karonen 2010-08-02T15:33:29Z 2010-08-02T15:33:29Z <p>I think it can. In particular, Wikipedia's <a href="http://en.wikipedia.org/wiki/Antiderivative#Antiderivatives_of_non-continuous_functions" rel="nofollow"><em>antiderivative</em></a> page seems to imply that, for any meagre $F_\sigma$ subset of an open interval, one can construct a function whose derivative is discontinuous on that subset. It also gives some examples.</p> http://mathoverflow.net/questions/34264/examples-of-badly-behaved-derivatives/34275#34275 Answer by Pete L. Clark for Examples of badly behaved derivatives Pete L. Clark 2010-08-02T16:01:45Z 2010-08-02T16:01:45Z <p>Of course every continuous function is a derivative, so your question seems like it could be reinterpreted as the <em>characterization problem for derivatives</em>: that is, give "nice" necessary and sufficient conditions on a function $f: [a,b] \rightarrow \mathbb{R}$ for it to be the derivative of some other function (one-sided, at the endpoints). </p> <p>As others have pointed out, any derivative must be a <strong>Darboux function</strong> -- i.e., satisfy the conclusion of the Intermediate Value Theorem -- and also a <strong>Baire class one function</strong> -- i.e., a pointwise limit of continuous functions. The latter implies that the set of discontinuities is meager. A lot of work has been done seeking to understand "how much bigger" the class of Darboux, Baire Class 1 functions is than the class of derivatives.</p> <p>Here are some references:</p> <blockquote> <p>Bruckner, A. M.; Leonard, J. L. <em>Derivatives</em>. Amer. Math. Monthly 73 1966 no. 4, part II, 24--56. </p> </blockquote> <p><a href="http://www.math.uga.edu/~pete/BrucknerLeonard66.pdf" rel="nofollow">http://www.math.uga.edu/~pete/BrucknerLeonard66.pdf</a></p> <blockquote> <p>Bruckner, Andrew M. Differentiation of real functions. Lecture Notes in Mathematics, 659. Springer, Berlin, 1978. x+247 pp.</p> </blockquote> <p><a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.bams/1183545222" rel="nofollow">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.bams/1183545222</a></p> <blockquote> <p>Bruckner, Andrew M. The problem of characterizing derivatives revisited. Real Anal. Exchange 21 (1995/96), no. 1, 112--133. </p> </blockquote> http://mathoverflow.net/questions/34264/examples-of-badly-behaved-derivatives/34277#34277 Answer by Pietro Majer for Examples of badly behaved derivatives Pietro Majer 2010-08-02T16:28:28Z 2010-08-02T16:35:59Z <p>Talking about how bad may be the derivative of an everywhere derivable function on the interval [a,b], the natural example that occurs to my mind is: a Pompeiu derivative, that is, a derivative that vanishes in a dense set (these weird functions, however, constitue a closed linear space of $C^0[a,b]$, while it's not even obvious that they are closed wrto addition!). For an account on the subject you may refer e.g. to the above quoted book by Andrew M. Bruckner <em>Differentiation of real functions</em>: it's also in the CRM series now (here is a preview: <a href="http://books.google.it/books?id=fXfEG-F2zJUC&amp;printsec=frontcover&amp;source=gbs_book_other_versions#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.it/books?id=fXfEG-F2zJUC&amp;printsec=frontcover&amp;source=gbs_book_other_versions#v=onepage&amp;q&amp;f=false</a>)</p> <p>For a quick reference, you may also have a look to the wiki article <a href="http://en.wikipedia.org/wiki/Pompeiu_derivative" rel="nofollow">http://en.wikipedia.org/wiki/Pompeiu_derivative</a> (it's me who wrote it ;-) )</p> http://mathoverflow.net/questions/34264/examples-of-badly-behaved-derivatives/34681#34681 Answer by Greg Marks for Examples of badly behaved derivatives Greg Marks 2010-08-05T20:24:18Z 2010-08-05T20:24:18Z <p>In Y. Katznelson and K. Stromberg&#8217;s paper &#8220;Everywhere differentiable, nowhere monotone, functions,&#8221; <i>Amer. Math. Monthly</i> <b>81</b> (1974), no. 4, 349&#8211;354, there is a construction, based on somewhat similar ideas to those described in Majer&#8217;s above mentioned article, whereby for any two disjoint countable sets $A, B \subset \mathbb{R}$ there exists a differentiable function whose derivative equals $1$ on $A$ and is less than $1$ on $B$.&#160; So if $A$ and $B$ are both dense in $\mathbb{R}$ then this derivative must be discontinuous at every point of $B$.</p> http://mathoverflow.net/questions/34264/examples-of-badly-behaved-derivatives/83333#83333 Answer by Joseph Pedersen for Examples of badly behaved derivatives Joseph Pedersen 2011-12-13T14:08:31Z 2011-12-13T14:08:31Z <p>I am wondering the same thing, about 'how' discontinuous a derivative of a function from R to R can be. Specifically, can a derivative be nowhere continuous?</p> <p>Consider the function defined as: F_a(x) = (x-a)^2 sin(1/(x-a)) , for x &lt;> a; 0 , for x=a.</p> <p>It is differentiable everywhere, and it's derivate at x=a is zero, but it's derivative is discontinuous at x=a.</p> <p>Now let G(x) = the sum from n=1 to inf of (1/2)^n F_[(1/2)^n].</p> <p>This should be a well defined function, which is everywhere differentiable, but whose derivative is discontinuous at x = (1/2)^n for all n in Z.</p> <p>Next, given that the rationals are countable, there exists a bijection b: Z+ to Q.</p> <p>Now let H(x) = the sum from n=1 to inf of 1/2^n F_[b(n)].</p> <p>This should be a well defined function (it's existence I mean, I don't suggest that it is constructible), which is everywhere differentiable. I am not sure if it can be proven that its derivative is discontinuous on Q?</p> <p>Anyway, I would be very excited to see a proof of either the existence or impossibility of an everywhere differentiable function whose derivative is everywhere discontinuous!</p> <p>Joe</p>