Let $f\colon X\to Y$ be a surjective continuous map between two topological spaces such that $X,Y$ are path-connected and such that every fibre $f^{-1}(y)$ is connected, for each $y\in Y$. Is there always a continuous section $s\colon Y\to X$ (i.e. a continuous map $s$ such that $f\circ s$ is the identity on $X$)?
If no, what kind of sufficient conditions could we impose on $f$ ? For example, is it better if $Y$ is supposed to be simply connected? Or if we choose $Y$ to be the interval $[0,1]$ ?



Existence of a continuous section
No, even if $Y=[0,1]$. The piecewise linear continuous nondecreasing surjection $f:[0,1] \rightarrow [0,1]$ which maps $[1/3,2/3]$ to $1/2$ and is otherwise 1-1 and linear has no continuous section.


