TITLE: Can the composition of two non-uniformly continuous functions be uniformly continuous?
QUESTION [2 upvotes]: I know that the composite function of two uniformly continuous functions is also uniformly continuous. But can the composition of two functions be uniformly continuous even when the functions themselves are not? What if one of them is allowed to be uniformly continuous?

REPLY [1 votes]: This is sort of stolen from Mariano Suárez Álvarez's answer, but here both functions are non-uniformly continuous:
Let
$$ f(x)=\begin{cases}-x^3&\text{if }x\ge 0\\-\sqrt[3]x&\text{if }x\le 0\end{cases}$$
and compose it with itself.

REPLY [1 votes]: Consider $f(x)=\begin{cases}\pi&\text{if }x<-1\\ 0&\text{if }x\ge -1\end{cases}$. Then, $f\circ \sin=\sin\circ f=0$ and $f\circ f=0$.
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