We know that the set of continuous function on $\mathbb R^n$, $\mathcal{C}(\mathbb{R^n})$ is equipped with $||.||_{L^{\infty}(\mathbb R^n)}$,
where $||f||_{L^{\infty}(\mathbb{R^n})}:=sup|f(x)|$, s.t $x \in \mathbb{R^n}$.
My question is, suppose I have an arbitrary function $f:\mathbb{R^n} \rightarrow \mathbb{R}$, and I want to prove that $f$ is continuous on $\mathbb{R^n}$, Is it sufficient to only prove that $||f||_{L^{\infty}(\mathbb{R^n})}$ is finite? So that I get my request to prove?