Composition of continuous and bounded function is bounded?

Question

Suppose $f$ is a continuous function and $g$ is a bounded function. Is it true true that $f\circ g$ is bounded?

It is to show there exists some $M>0$ such that $\lvert f(g(x))\rvert\leq M$ for all $x$-

Answer 1

In general, no. Consider

\begin{alignat*}{3} g(x)&=\arctan x,&&x\in(-\infty,+\infty),\\ f(x)&=\log\left(\dfrac{\pi^2}{4}-x^2\right)\qquad && x\in(-\pi/2,\pi/2). \end{alignat*}

$g(x)$ is bounded, $f(x)$ is continuous on the given interval. However, $f(g(x))$ is unbounded on $(-\infty,+\infty)$.

Answer 2

Yes, if $|g(x)| \leq C$ for all $x$ then $f$ is bounded on the compact interval $[-C,C]$ so $f\circ g$ is bounded.