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Possible Duplicate:
Are Continuous Functions Always Differentiable?

If such a function exists, can anyone give an example of a function $f(x) : \mathbb{R} \longrightarrow \mathbb{R}$ that is continuous for all $x \in \mathbb{R}$ but differentiable nowhere?

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    Another non-duplicate, but whose answers answer this: http://math.stackexchange.com/questions/150/are-there-any-functions-that-are-always-continuous-yet-not-differentiable-or-v (to go along with http://math.stackexchange.com/questions/7923/are-continuous-functions-always-differentiable, pointed out by Moron). Actually, these two "duplicates" are really duplicates of each other, not of this one, but I have also voted to close.2011-04-05

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The Weierstrass function mentioned in Jesse Madnick's answer is the standard example, but I think this example is slightly misleading. The fact that it is constantly presented as the standard example may suggest that such examples are rare and must be constructed in a certain way. Actually such examples are extremely common; in an appropriate sense, the "generic" continuous function is nowhere differentiable.

To my mind, the point of the Weierstrass function as an example is really to hammer in the following points:

  • The uniform limit of continuous functions must be continuous, but
  • The uniform limit of differentiable functions need not be differentiable.

However, if $f_n(x)$ is a uniformly convergent sequence of differentiable functions such that the derivatives f_n'(x) also converge uniformly, then the uniform limit $f(x)$ is differentiable, and f'(x) is the uniform limit of the functions f_n'(x). So what fails in the example of the Weierstrass function is that the derivatives do not even come close to converging uniformly.

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    @mike4ty4 Actually, the set of nowhere differentiable functions is dense in the set of continuous functions (with respect to the uniform topology). See also this answer: http://math.stackexchange.com/a/165731/31482014-10-13
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Another popular example is what I know as Takagi's Function.

It is somehow different from the Weierstrass Function in that it is not constructed as a uniform limit of differentiable functions. However, it is a uniform limit of continuous functions in a way that the points of non-differentiability populate the "whole interval" (if that point of view makes any sense...).

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    this i think is good because its something thats easy to remember and draw pictures for (if giving it as an example to someone)2011-04-05
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A very famous example - and by far the most important when it comes to practical applications (finance: option pricing!) - is the Wiener process.

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See Wikipedia's page on the Weierstrass Function.