Does such a function even exist?

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Discussion Overview

The discussion revolves around the existence of functions that are continuous everywhere but differentiable nowhere. Participants explore various examples and definitions, including the Weierstrass function and other proposed functions, while also discussing related concepts such as fractals and the Mandelbrot set.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant recalls a function described by their father that resembles an infinitely small sawtooth and questions its definition.
  • Another participant mentions the Weierstrass function as an example of a function that is continuous everywhere and differentiable nowhere.
  • A participant proposes a function defined as f(x) = 1/x for rational x and f(x) = 0 for irrational x, claiming it is continuous everywhere but differentiable nowhere.
  • Several participants challenge the continuity of the function proposed by JonF, stating it is continuous only on the irrationals and discontinuous on the rationals.
  • There is a suggestion to explore fractals, with the Weierstrass function highlighted as a notable example.
  • One participant introduces a recursive sequence related to the Mandelbrot set, questioning its classification as a function.
  • Another participant acknowledges the Mandelbrot set's properties, noting it is not a function but is continuous everywhere and differentiable nowhere.
  • There is a discussion about the boundary of the Mandelbrot set and whether it can be parameterized, with uncertainty expressed about the existence of such a parameterization.
  • A participant speculates that if a smooth parameterization of the boundary exists, it would have to have a Hausdorff dimension of 1.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the examples of functions that are continuous everywhere and differentiable nowhere. The discussion remains unresolved, with differing opinions on the continuity of proposed functions and the classification of the Mandelbrot set.

Contextual Notes

Some claims about continuity and differentiability depend on specific definitions and interpretations, which are not universally agreed upon in the discussion. The relationship between fractals and functions is also a point of contention.

Izzhov
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When my father was in college, one of his professors showed him a function that is continuous everywhere and differentiable nowhere. All my dad remembers is that the function was like an infinitely small sawtooth. If this function exists, how can it be defined?
 
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i think this function is continuous everywhere:

f(x) = 1/x if x is rational
f(x) = 0 if x is irrational

but differentiable no where
 
The function given by JonF is continuous on the irrationals, and discontinuous on the rationals. So it is not continuous everywhere.

Perhaps Izzhov would enjoy the text 'Counterexamples in Analysis".
 
OP: Try searching up fractals. The Weierstrass function is a good example, and somewhat the first one found.
 
The function given by JonF is continuous on the irrationals, and discontinuous on the rationals. So it is not continuous everywhere.

Doesn't look continuous anywhere.
 
jostpuur said:
Doesn't look continuous anywhere.

Indeed you are right.
However, the function:
[tex]f(x)=\frac{1}{q}, x=\frac{p}{q}, p,q\in\mathbb{N}[/tex]
and f(x) zero elsewhere IS continuous on the irrationals, but not on the rationals.
 
JonF, you might want to reconsider if 0 is a rational number >.<
 
  • #10
Would this be a simpler example?

[tex]x_n=(x_{n-1})^2 + C[/tex]
[tex]x_0=C[/tex].

I think I saw something like this in the favorite equations thread, general math section.
 
  • #11
In what sense is that a function?
 
  • #12
I think he considers Mandelbrot set a "function" .
(http://en.wikipedia.org/wiki/Mandelbrot_set) .
:smile: :smile: :zzz:

It is a beautiful "monster" Gib_Z,but far from being function.
 
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  • #13
O i know its not a function, but I thought it was continuous everywhere and differentiable nowhere, its a fractal right? I know very little about this area.
 
  • #14
i know its not a function, but I thought it was continuous everywhere and differentiable nowhere
A continuous set? :rolleyes: I would guess, that you are talking about the boundary of the Mandelbrot set? It looks bad enough, that if you had a parametrisation for it, then it would not be differentiable. But at this point I must raise a different question: Is there parametrisation for boundary of Mandelbrot set? I have a feel that there's not, but I'm not sure.
 
  • #15
I would imagine that if there was a smooth parameterization of the boundary then it would have to have Hausdorff dimension 1.
 

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