I'm studying uniform continuity in metric spaces.
Let ($X$,$d$), ($Y$,$d'$) be metric spaces, $f:X \to Y$.$f$ is said to be uniformily continuous on $X$ iff $$ \forall \epsilon > 0 \; \exists \delta( \epsilon) >0 \; \forall x,x' \in X : d(x,x')< \delta \Rightarrow d'(f(x), f(x')) < \epsilon$$
And as an example we prove that $\sin(x)$ is uniformily continuous on $\mathbb R$, with the usual metric, and many more. My question is:
Is it true that $\sin(f(x))$ is uniformily continuous in some subset of $\mathbb R$ iff f(x) is uniformily continuous on some subset of $\mathbb R$?
I feel that this isn't true, but I don't have any idea how to prove this.
