should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that $f(x)=x^2,f:\mathbb{C}\longmapsto\mathbb{C} $ is not uniformly continuous on whole $\mathbb{C}$, but $f(x)$ is uniformly continuous if $\mid{x}\mid<20$
I struggle here, since we just used it in $\mathbb{R}$ , but not in $\mathbb{C}$
Thank you
You can make the definition of uniform continuity you have work in $\mathbb C$. Specifically, a function $f:\mathbb C \to \mathbb C$ is uniformly continuous if for every $\epsilon>0$ there exists a $\delta>0$ such that for all $x,y\in \mathbb C$ with $|x-y|<\delta$, $|f(x)-f(y)|<\epsilon$. Essentially, you are now working with quanities in $\mathbb R$, and you can proceed as normal.