Use the following theorem:
"A function that is analytic in a domain $D$ is uniquely determined over $D$ by its values in a domain, or along a line segment, contained in D"
to show that if $f(z)$ is analytic and non-constant throughout a domain $D$, then it cannot be constant throughout any neighborhood lying in $D$.
I am given the following suggestion: Suppose that $f(z)$ does have a constant value $w_0$ throughout some neighborhood in $D$.
My definition of analytic is: a function $f$ of a complex variable is said to be analytic at a point $z_0 \in C$ if f is differentiable in a neighborhood of $z_0$
I am in a complex analysis class. I have to prove this. I am thinking:
Assume that there exists an open set $U \subseteq D$ and $w_0 \in C$ such that $f(z)= w_0$ for all $z \in U$.
then by the theorem above $f(z)$ is not constant throughout since there exists a unique subset of D that is not constant.
Am I wording this okay? Is what I have written sufficient?