Let $ f : U \subset \mathbb C \to \mathbb C $ a continuous function where $ U $ is an open and connected set.
Prove that the image of $ f$ cannot be a curve on compex plane.
Let $ f : U \subset \mathbb C \to \mathbb C $ a continuous function where $ U $ is an open and connected set.
Prove that the image of $ f$ cannot be a curve on compex plane.
Are you sure there are no additional constraints, e.g. the Cauchy-Riemann equations are satisfied (continuity + C.R. -> analytic). One could simply take $f(z) = \Re(z)$, which is continuous but not analytic -- the image of an open set (say the ball of radius 1 around the origin) is a curve (the interval $[-1,1]$).
The reason this cannot hold for an analytic function is because of the open mapping theorem. Analytic functions are open mappings, so an open set $U$ has to map to an open set $f(U)$, which cannot be a curve.