Since we can represent complex functions as $f(z) = u(x,y) + iv(x,y)$, under what conditions do we know that $f(z)$ is continuous? For example, if $u(x,y)$ and $v(x,y)$ are both continuous can we conclude that $f(z)$ is continuous? Does anyone know of a proof of this, or just an intuitive explanation of why or why not this is true? Furthermore, if either $u(x,y)$ or $v(x,y)$ are not continuous does this imply that $f(z)$ is not continuous? This isn't for a particular question I'm just wondering about how arguments involving continuity transfer over to complex valued functions.
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The key here is that topologically $\mathbb{C}$ and $\mathbb{R}^2$ are the same thing. Namely, it really behooves us to think of functions $f(x,y)$ which are mappings $\mathbb{R}^2\to\mathbb{R}^2$. Then, it is a common fact (because $\mathbb{R}^2$ is given the so-called product topology) that a map $X\to\mathbb{R}^2$ will be continuous if and only if its two coordinate functions are continuous. Your $u$ and $v$ are precisely the coordinate functions of $f$. |
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