Continuity of partials of real and imaginary parts of holomorphic function

Question

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.

Answer 1

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.