One of the first examples of a (real smooth) non holomorphic function on $\mathbb{C}$ is complex conjugation: $f(z) = \overline{z}$ is never holomorphic. My question is the following.
Is there a holomorphic function $u(x+iy)$ defined in an open set containing the line $x = 0$ such that $u(0+iy) = -iy$?
My intuition says it is not true, it should not satisfy Cauchy Riemann as well, but I do not know how to prove this.