Is there such a thing as an uncountable polynomial?

[Number Nine]

That adds the requirement that function is differentiable everywhere. Pretty sure you've no chance of finding a function like that. Need to rephrase it without recourse to differentiation.

The only way I can think of, and it isn't very satisfying, is as follows: Does there exist a continuous, nowhere-constant function with uncountably many local extrema?

Elaborating on "nowhere-constant", let's say that a function f is constant at a point x if f assumes a single value within some neighbourhood of x.