I am currently reading Complex Analysis by Stein and found the follow theorem (Theorem 2.4 on Page 13):
Suppose that $f = u + iv$ is a complex-valued function defined on an open set $\Omega$. If $u$ and $v$ are continuously differentiable and satisfy the Cauchy-Riemann equations on $\Omega$, then $f$ is holomorphic on $\Omega$ and $f'(z) = \frac{ \partial f }{ \partial z }.$
But if $u$ is any continuously differentiable function on a disc and we define $v$ through the Cauchy-Riemann equations, then the complex-valued function given by $f = u + iv$ would be holomorphic by the theorem, which then implies that $u$ is indefinitely differentiable since $f$ is indefinitely differentiable and $u$ is the real part of $f$. This means that continuous differentiable implies indefinitely differentiable, which is obviously impossible.
What's wrong with my reasoning?
Thanks in advance.
Best Answer
What's specifically wrong is your use of the word define in this sentence: "we define $v$ through the Cauchy-Riemann equations".
CR is not a definition, it is a system of differential equations, which may not have a solution, as other comments and answers are pointing you to.