This is a strange question but I am interested to know in this problem.
Let $\Omega \subset\mathbb{R}^{N}$ be a bounded smooth domain. Assume $f : \overline{\Omega}\to\mathbb{R}$ is continuous. Then, we have the following definition : $$\forall x \in \overline{\Omega}, \forall \varepsilon > 0, \exists \delta(x,\varepsilon) > 0 \ni |x-y|<\delta(x,\varepsilon)\implies|f(x)-f(y)|<\varepsilon$$
Now, my question is, since $\overline{\Omega}$ is a compact set, can we always find $\delta(x,\varepsilon)>0$ so that it becomes a continuous function with respect to $x$ and $\varepsilon$?
For simple function like $y=mx+c$ in one-dimensional case for finite interval $[a,b]$ the question is trivial but I am not sure about $N$-dimensional case with arbitrary continuous function.