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Results tagged with real-analysis
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user 4354
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
36
votes
Accepted
Smooth functions for which $f(x)$ is rational if and only if $x$ is rational
There are such functions. Moreover any diffeomorphism $f_0:\mathbb R\to\mathbb R$ can be approximated by such $f$. For the sake of simplicity I assume that $f_0'\ge 2$ everywhere.
Enumerate the ratio …
- 32.4k
15
votes
Accepted
A generalization of intermediate value theorem on R^k
The statement is true. It is almost precisely Lemma 2 in the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210. The proof is short but not easy to invent. The paper can be read …
- 32.4k
4
votes
Accepted
Estimating the Hausdorff measure of a subset of the sphere
Yes. Here is an elementary proof.
Construct a distance non-increasing retraction $p:S^{n-1}\setminus B_r(a)\setminus B_r(b)\to \partial B_r(a)$. (For example: divide the sphere into two-hemispheres b …
- 32.4k
29
votes
Why is Lebesgue integration taught using positive and negative parts of functions?
The integration of nonnegative functions deserves its own chapter, just like nonnegative measures. It has more features than the general case and there are cases when you need exactly these features a …
- 32.4k
122
votes
Accepted
solving $f(f(x))=g(x)$
Q1: No. Let $g(0)=1, g(1)=0$ and $g(x)=x$ for all $x\in\mathbb R\setminus\{0,1\}$.
Assuming $f\circ f=g$, let $a=f(0)$, then $f(a)=1$ and $f(1)=g(a)=a$ since $a\notin\{0,1\}$.
Then $g(1)=f(f(1))=f(a)= …
- 32.4k