Reformulate continuity more rigorously

Question

In a textbook I am reading we show the continuity of a function $f\colon V \rightarrow L^{\infty}(\mathbb{H})$, with $f(0)=0$ and where $(V,\|\cdot\|)$ is a vector space together with a norm $\|\cdot\|$ and $L^{\infty}(\mathbb{H})$ with norm $\|\cdot\|_{\infty}$. The authors justify its continuity by saying that if we change $v\in V$ by a small amount then the function $f(v) \in L^{\infty}(\mathbb{H})$ also changes by a small amount. How can I formulate or translate this into a more mathematically rigorous proof for example with the $\epsilon$-$\delta$-notation?

Answer 1

$f$ is continuous at $v$ if, for each $\epsilon>0$ there exists some $\delta >0$ such that, for all $u \in V$ with $\| u-v \| <\delta$ we have $$\| f(u)-f(v) \|_\infty < \epsilon$$