Is inequality in the norm of $L^{\infty}$ holds for all points?

Question

Let $f\in L^{\infty}(\mathbb{R}^n)$. It is well known from functional analysis that $ f(x) \leq ||f(x)||_{\infty} $ for almost everywhere $x\in \mathbb{R}^n.$ I suspect that if $f$ is continuous, this inequality holds for all points $x\in \mathbb{R}^n$, precisely, since locally f is bounded. Is this true? How can we justify?