Question about the C-R equation and the Analytic functions

Question

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.

Answer 1

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.

Answer 2

A necessary condition for a real valued function $u$ to be the real part of an analytic function is that it is harmonic. Not every continuously differentiable function is harmonic, for example $f(x,y)=x^2$ is not. Therefore, no matter how you will define $v$, you will never get an analytic function if you start with non-harmonic $u$.

For more information, you may want to consult the first answer here.