PROBLEM
Let $G\subset\mathbb{C}$ be a region (open and connected) and let $f:G\rightarrow\mathbb{C}$ be a holomorphic function such that $f'(z)=0$ for all $z\in G$. Prove that $f$ is constant on $G$.
PROOF
Let $G,f$ be as above. Let $z_0\in G$ and define $S:=\{z\in G: f(z)=f(z_0)$. By continuity of $f$ we have that $S$ is closed. Let $a\in S$. Then there exists $\epsilon>0$ such that $B(a,\epsilon)\subset G$. Let $z$ be in this ball and define $g(t):=f(tz+(1-t)a)$ for $0\leq t \leq 1$. Using the chain rule we have that $g'(t)=0$ for $0
Let $G\subset\mathbb{C}$ be a region (open and connected) and let $f:G\rightarrow\mathbb{C}$ be a holomorphic function such that $f'(z)=0$ for all $z\in G$. Prove that $f$ is constant on $G$.
PROOF
Let $G,f$ be as above. Let $z_0\in G$ and define $S:=\{z\in G: f(z)=f(z_0)$. By continuity of $f$ we have that $S$ is closed. Let $a\in S$. Then there exists $\epsilon>0$ such that $B(a,\epsilon)\subset G$. Let $z$ be in this ball and define $g(t):=f(tz+(1-t)a)$ for $0\leq t \leq 1$. Using the chain rule we have that $g'(t)=0$ for $0
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