I'm currently doing a course on the theory of metric spaces.
This is the version of Schauder fixed point theorem from my course:
Let $(X,\|\cdot\|)$ be Banach and $C\subset X$ a closed, bounded, convex subset. If $ f\colon C \to C $ is a compact map, then there is a $x_*\in C$ such that $f(x_*)=x_*$.
Recall:
A continuous map $f\colon C\to C$ (where $C\subset X$) is compact if for any bounded sequence $(c_n)$ in $C$ the sequence $\{f(c_n)\}$ has a convergent sub-sequence.
This is my question:
If I do not demand that $f$ is continuous, (i.e. I call a map $f\colon C\to C$ compact if for any bounded sequence $(c_n)$ in $C$ the sequence $\{f(c_n)\}$ has a convergent sub-sequence), does (my version of) the Schauder fixed point theorem still hold?