$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema? I couldnt think of any; by intuition I  don't think any can exist, but I can't figure out how to prove it.  If it existed then the set of extrema would have to be uncountable but I think this might somehow violate continuity.

Take a closed uncountable set without interior points, such as the Cantor set $C$. The distance function 
$$f(x) = \min\{|x-y|:y\in C\}$$
is continuous, not constant on any interval, and has a minimum at every point of $C$. 
