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pow:problem4s21

Problem 4 (due Monday, April 12)

a) Let $f:\mathbb R \longrightarrow \mathbb R$ be a differentiable function such that $f(\sin x)=\sin f(x)$ for every $x\in \mathbb R$. Prove that if $f$ is not identically zero then $\displaystyle \lim_{x\to 0} \frac{f(x)}{x}$ exists and is equal to $1$ or $-1$.

b) Prove that there is a continuous function $f:\mathbb R \longrightarrow \mathbb R$ such that $f(\sin x)=\sin f(x)$ and $\displaystyle \lim_{x\to 0^+} \frac{f(x)}{x}$ does not exist.

Two solutions were submitted: by Paul Barber and Ashton Keith. Neither one is complete. Ashton attempts to solve part a) under additional assumption that $f'$ is continuous at 0. While his solution has some gaps, the ideas are very nice indeed and they can be improved to a complete solution (under the additional assumption). For more details and to see complete solutions see the following link Solution.