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.