Example of a Function

  • #1
Can someone give me an example of a function that is continuous everywhere yet differentiable nowhere?
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
Have you seen the Koch Snowflake, or any other fractal curve?
 
  • #3
Seen them yes. Unfortunately I haven't studied them. I'm told that the discovery of such functions is what prompted the beginning of mathematical analysis. How is any such function defined?
 
  • #4
See Mathworld, Planetmath and Wikipedia. Fractal curves are usually defined recursively. Details of each curve can be found in their respective articles.
The Weierstrass function is a related pathology. Weierstrass was one of the mathematicians (including Dedekind, Cantor, Kronecker, and so forth) that heralded the 2nd age of rigor, which put on firm ground many concepts that were previously nebulous in definition and application. Many pathologies were created at this time.
 
  • #5
benorin
Homework Helper
Insights Author
Gold Member
1,274
80
Here's a function which is everywhere continuous but nowhere differentiable

[tex]\forall x\in\mathh{R}\mbox{ let } f_{0}(x)[/tex] be the distance from x to the nearest integer; thus [tex] f_{0}(x+1)=f_{0}(x)\mbox{ and }f_{0}(x)=\left| x \right|,\mbox{ for }\left| x \right| \leq \frac{1}{2}[/tex].
For each [tex]n\in\mathbb{N}[/tex], define [tex]f_{n}(x)=\frac{f_{0}(12^{n}x)}{2^{n}}[/tex], then [tex]f(x)=\sum_{n=0}^{\infty} f_{n}(x)[/tex] is such a function. (continuity is from uniform convergence)
 
  • #6
quasar987
Science Advisor
Homework Helper
Gold Member
4,778
11
hypermorphism said:
See Mathworld, Planetmath and Wikipedia. Fractal curves are usually defined recursively. Details of each curve can be found in their respective articles.
The Weierstrass function is a related pathology. Weierstrass was one of the mathematicians (including Dedekind, Cantor, Kronecker, and so forth) that heralded the 2nd age of rigor, which put on firm ground many concepts that were previously nebulous in definition and application. Many pathologies were created at this time.
This might not be the most on-topic post ever but lookit it's pretty funny:

My applied analysis textbook cover said:
Mr.Cauchy anounces that, in order to conform to the will of the Council, he will not anymore take care of providing, as he has done up until now, perfectly rigourous proofs of his statements.
Council of instruction of l'École Polytechnique,
november 24 1825
:rofl: :rofl: :rofl: :rofl:
 
  • #7
486
0
i have a stupid question
Can you even take derivative if it is not continuous?
Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.
 
  • #8
NateTG
Science Advisor
Homework Helper
2,450
5
leon1127 said:
i have a stupid question
Can you even take derivative if it is not continuous?
Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.
There are functions that have the same derivative on both sides of discontinuities (like the greatest integer function), but the derivative at the discontinuity is undefined.
 
  • #9
benorin
Homework Helper
Insights Author
Gold Member
1,274
80
Since there are everywhere continuous but nowhere differentiable functions, but every continuous function is integrable...[end my rant because Mathworld said it better, see below].

...continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

The above quote is from http://mathworld.wolfram.com/Derivative.html

So in what "sense" is that [the " "more" " part]? I asked my Real analysis prof, he said in the sense of catagory. But I wonder, could it be true in the sense of cardinality? Specifically, is the cardinality of the set of all functions which are integrable greater than that of differentiable functions?
 
  • #10
benorin
Homework Helper
Insights Author
Gold Member
1,274
80
leon1127 said:
i have a stupid question
Can you even take derivative if it is not continuous?
Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.
Continuity is a neccessary, but not a sufficient condition for differentability. So no, you cannot differentiate a non-continuous function.
 
Last edited:
  • #11
486
0
benorin said:
Continuity is a neccessary, but not a sufficient condition for differentability. So no, you cannot differentiate a non-continuous function.
So it means that suck function doesnt exist?
 
  • #13
665
0
I think he meant "such" function.
 
  • #14
740
13
lmao :rofl:
 
Last edited:
  • #15
If I ever create an original pathology, its name shall be suck. :rofl:
 
  • #16
218
0
Treadstone 71 said:
Can someone give me an example of a function that is continuous everywhere yet differentiable nowhere?
Well it's technically not a function, but how about 0y+3x=6
it's a straight line, with an infinite value both plus and minus for every x different than 2.

Nah...that's not what you're looking for...what about f(x)=0?
I don't know...i tried.
 
  • #17
HallsofIvy
Science Advisor
Homework Helper
41,833
956
leon1127 said:
i have a stupid question
Can you even take derivative if it is not continuous?
Assume it is not continuous at point C, then the limit of the slope approve to C from both sides should not be equal, thus derivative does not exist and nondifferentiable.
Just about every calculus book has a proof that a function is differentiable at c only if it is continuous at c. There is no such function.
 

Related Threads on Example of a Function

  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
2
Views
3K
Replies
1
Views
2K
Replies
4
Views
2K
  • Last Post
Replies
7
Views
2K
Replies
4
Views
1K
Top