Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.

Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then $\text{Re}(f(z))^{-1}(c)$ is an union of differentiable curves in the plane.

** Question: **

If $c$ is not a regular value and $\text{Re}(f(z))^{-1}(c)$ have at least one cluster point is this set also a piece-wise differentiable curve ?