Is there such a thing as an uncountable polynomial?

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Discussion Overview

The discussion revolves around the concept of whether a polynomial can possess an uncountable number of "turns," interpreted as local minima and maxima. Participants explore the implications of polynomial properties, the nature of functions beyond polynomials, and the definitions of critical points in the context of various mathematical functions.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants assert that a polynomial must have a finite number of "turns," as the derivative of a polynomial is also a polynomial with a finite number of roots.
  • Others suggest that while polynomials are limited in their number of critical points, other functions, such as sine and cosine, can have an infinite number of local minima and maxima.
  • A participant introduces the Weierstrass function, questioning its properties in relation to local minima, but acknowledges that it is not a polynomial.
  • Some participants discuss the implications of the Weierstrass function's differentiability and its potential lack of minima/maxima based on the definitions being used.
  • There is a suggestion that the original question may be better framed in terms of continuous functions rather than polynomials specifically.
  • One participant proposes that the definition of minima and maxima may need to be clarified, particularly in the context of functions that are not polynomials.

Areas of Agreement / Disagreement

Participants generally agree that polynomials cannot have an uncountable number of turns, but there is disagreement regarding the properties of other types of functions and whether they can exhibit such behavior. The discussion remains unresolved regarding the broader implications for continuous functions.

Contextual Notes

Participants note that the definitions of minima and maxima may vary, particularly when considering functions that are not polynomials. The discussion also highlights the limitations of applying polynomial properties to other types of functions.

  • #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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