Show that holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}$ is constant Let's $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function such that values $f$ are on line $y=ax+b$. Show that $f$ is constant.
I think I should use Cauchy-Riemann equations but I don't know what does mean that values $f$ are on line $y=ax+b$. Can you explain me that?
Thanks in advance.

I'll assume you meant $x$ and $y$ are the real and imaginary parts of $f(z)$ and $a$ and $b$ are real.
Suppose $f'(z_0)\ne 0$.  Choose $\varepsilon$ so close to $0$ that when $|\Delta z|=\varepsilon$ then $f(z_0+\Delta z)$ differs from $f(z_0)+f'(z_0)\,\Delta z$ by less than $\Delta z/1000000$.  Then as $\Delta z$ goes around the circle $\{z:|z-z_0|=\varepsilon\}$, then $f(z_0)+f'(z_0)\,\Delta z$ goes around the circle $\{w:|w-f(z_0)| = |f'(z_0)|\varepsilon\}$.  That circle is not on a straight line, and $f(z)$ stays close to $f(z_0)+f'(z_0)\,\Delta z$.
