Let $X$ be a set, let $f:X\to\mathbb R$ be a [bounded function](/page/Bounded%20Function), and let $I\subset\mathbb R$ be a bounded interval such that $f(X)\subset I$. If $\varphi:I\to\mathbb R$ is bounded, then the composition $\varphi\circ f:X\to\mathbb R$ is bounded.