Define $C([a,b], \mathbb R)$ to be the space of continuous functions $f : [a,b] \to \mathbb R$ with the norm $\| \cdot \|_{\infty}$. Let $H : C([a, b], \mathbb{R}) \rightarrow \mathbb{R}$ be the map such that $H(f)$ is the definite integral of $f$ from $a$ to $b$ for any $f \in C([a,b], \mathbb R)$.
How do I prove that $H$ is continuous?
Note [SN]: Edited the question to make it more complete.