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I have repeatedly come across the following sort of argument:

Let $f = u+iv$ be a holomorphic function. Then since it is infinitely differentiable, so are $u$ and $v$. Thus, the partial derivatives of $u$ and $v$ exist and are continuous.

I not sure why these partials must be continuous. In fact, I'm not totally sure what it means for $u$ and $v$ to be infinitely differentiable as functions $\mathbb{R}^2 \to \mathbb{R}$ in this context.

I would appreciate some clarification.

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Although this is not the definition of $C^\infty$ function (you can see the definition here), a function $f\colon\mathbb R^n\longrightarrow\mathbb R$ is a $C^\infty$ function if and only if it has partial derivatives of all orders and they are continuous functions. So, the theorem that you mentioned states that the real part and the imaginary part of a holomorphic function have this property.

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  • $\begingroup$ This is very helpful. I have only ever seen $C^{\infty}$ deined in terms of the existence/continuity of all partials. Is the equivalence you mention a standard result of basic real analysis? I just wonder if the authors of the texts I was reading (which were pretty basic) had a different idea in mind. $\endgroup$ Sep 10 '19 at 4:17

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