Local Extrema of Functions Exhibiting Self-Similarity

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Discussion Overview

The discussion revolves around the properties of the Weierstrass function, particularly regarding the nature of local extrema in the context of its self-similarity. Participants explore the implications of the function being continuous everywhere but differentiable nowhere, questioning whether every point can be classified as a local extremum.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions if every point of the Weierstrass function can be a local extremum, given its self-similarity and the definition of local extrema.
  • Another participant emphasizes the importance of the definition of extremum, which involves a neighborhood, and challenges the feasibility of every point being an extremum.
  • A further response suggests that logically, every point cannot be an extremum, but raises the question of what constitutes a "neighborhood" in this context.
  • Another participant asserts that the definition from Calculus implies that every point cannot be an extremum, as neighborhoods must contain other points that are not extremal.

Areas of Agreement / Disagreement

Participants generally agree that not every point can be a local extremum, but there is some contention regarding the definition and implications of "neighborhood" in this context.

Contextual Notes

The discussion highlights the dependence on definitions of neighborhoods and local extrema, which may vary based on course material or interpretations.

moonlitdawn
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Hi guys. Today was my last day of Calculus I class, and I’ve had this question simmering in my mind for some time now. Perhaps you guys may enlighten me or point me in the right direction?

Consider the Weierstrass function, which is continuous everywhere but differentiable nowhere. We know that a local extremum, let’s say a local maximum, means that there is a number b, a < b < c, such that f(b) >= f(x) in (a, c). We also know that a local extremum either has a derivative of 0 or is non-differentiable.

My question is: Is it possible for every point of the Weierstrass function to be a local extremum (considering it is a non-constant function), given its self-similarity? If you look at it from one viewing rectangle it appears as if it the function has a maximum/minimum at a point in a given neighborhood, but then you zoom in and zoom in and zoom in. . . .

If not, how does one locate the points in which there is a local extremum?

Sorry if that was a lame question, and thanks in advance!
 
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You should recall the definition of extremum. It involves a neighborhood. Can every point be an extremum given this definition?
 
Yes, it involves a neighborhood. Well, logically the answer would be no, right? But what counts as a "neighborhood" here?
 
Well, use whatever definition you were given in your course of Calculus. Anyway it will have more than one point. So every point cannot be an extremum, because its neighborhood must contain other points, which are non-extremal.
 

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