What is an example of a function continuous at point but discontinuous on any $\epsilon$-balls?

Question

Let $E$ be open in $\mathbb{K}^n$ and $V$ be a normed space over $\mathbb{K}$.

Let $f:E\rightarrow V$ be a function which is continuous at a point $p\in E$.

Then, is it possible that for any $\epsilon>0$, $f$ is discontinuous on $B(p,\epsilon)$?

Answer 1

I assume that by $\mathbb{K}$ you mean a valued field that contains $\mathbb{Q}$ as a dense subfield, e.g. $\mathbb{R}$ or $\mathbb{C}$.

Let $f:E\to V$ be bounded and discontinuous everywhere, e.g. the characteristic function of $\mathbb{Q}^n\cap E$. Then $g(x)=f(x)\cdot |x-p|$ for $x\in E$ is continuous in $p$, but nowhere else.