Please give an example of a continuous function $f:[0,1]\to\mathbb R$ which doesn't have a derivative at any point. I can't think of anything, can someone help please?
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Most Brownian motion paths are like that. – kimchi lover Dec 02 '17 at 14:42
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Possible duplicate of Function example? Continuous everywhere, differentiable nowhere – Moishe Kohan Dec 03 '17 at 03:23
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This was asked so many times, already the example I gave was closed as a duplicate. – Moishe Kohan Dec 03 '17 at 03:23
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There is a well-known example of a continuous function that is not differentiable by Weierstrass:
Clashsoft
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Pavel Ievlev
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For example, $f(x)=1$ when $x$ is rational, and $f(x)=0$ when $x$ is irrational.
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Thats right, but I forget to write in condition that I need a continious function. – Mathworld Dec 02 '17 at 14:57
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Just take any function which is nowhere continuous.
Finding an $f$ which is continuous but nowhere differentiable is much harder. The most famous example is the Weierstraß function, which was explicitely constructed to have that property.
Before its construction, it was conjectured, that every continuous function was differentiable except on a set of isolated points.
So its not surprising that you cannot think of anything
user509325
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