If $f$ is holomorphic function then are the zeros of $\Re(f)$ isolated?

Question

Let $f$ be a holomorphic function.

We know that the the zeros of $f$ are isolated

I would like to know if is it true that the zeros of $\Re(f)$ are isolated.

Thanks.

Answer 1

Hint: consider $f(z) = z$ and the set $\{ z \in \mathbb{C} : \Re(z) = 0 \}$.

In fact, zeros of $\Re(f)$ are rarely isolated, because if $f(x+iy) = u(x, y) + i v(x, y)$ is holomorphic (where $u$ and $v$ are real), then under some natural assumption* the solution to the equation $u(x, y) = 0$ is a one-dimensional submanifold of $\mathbb{R}$, which is clearly not a discrete set.

*When $0$ is a regular value of $u(x, y)$ and $0 \in \operatorname{im} u$.