We have a function $f:U\rightarrow\mathbb{R}^m$ defined as $f(x)=(f_1(x),f_2(x),...,f_m(x))$ that for every $i$, $f_i:U\rightarrow \mathbb{R}$ is a $C^\infty$ function if every partial differential of $f_i$ is differentiable of every order in every points of $U$.
Now my question is:
a) Let $U\subseteq\mathbb{C}$ be open and $f:U\rightarrow \mathbb{C}$ analytic, Prove that $f$ is $C^\infty$.
So, what is the relationship between analytic and $C^\infty$ is this question?
How can I prove (a)?