Is there such a thing as an uncountable polynomial?

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SUMMARY

Polynomials cannot have an uncountable number of turns, as they are defined to have a finite degree, which limits the number of critical points (local minima and maxima) to a finite count. The derivative of a polynomial is also a polynomial, which has only a finite number of roots. While functions like sine and cosine can have infinite turns, they do not qualify as polynomials. The Weierstrass function, although it has complex properties, does not serve as a counterexample since it is not a polynomial and does not exhibit uncountably many local extrema.

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  • #31
haruspex said:
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.
 

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