Let $X$ be a compact Haussdorf topological space with the following property:

For every continuous function $f:X\to \mathbb{C}$ the image $f(X)$ is a payhconnected subset of $\mathbb{C}$.

>Is such a space $X$ necessarilly a path connected space?


This question is inspired by (and realated to) this post:

https://mathoverflow.net/questions/195900/unital-c-algebras-whose-all-elements-have-path-connected-spectrum