# What does it mean for a function $\phi$ to be “bounded on the real line”?

I have a problem that states

'$\phi(x)$ is a bounded function on the real line, $x\in \Bbb R$'.

Does this restrict $x$ to some domain $\Omega \subset \Bbb R$?

Or does this make $\phi(x) : \Bbb R \rightarrow \Omega \subset \Bbb R$ such that $sup(\Omega),inf(\Omega) \neq \infty$?

• The latter. Its range $f(\mathbb{R})$ must be a bounded set. – dxiv Oct 13 '16 at 22:44

It means there exists some constant $C \gt 0$ such that $|\phi(x)| \le C$ for all $x\in \mathbb{R}$.