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Are continuous functions always differentiable?

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Think of $f(x)=\left\vert x\right\vert $. –  Américo Tavares Oct 26 '10 at 15:15
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Out of curiosity, why does this have 4 upvotes and no downvotes? What makes it a good question? –  Pete L. Clark Oct 27 '10 at 7:49
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@Pete: questions get upvotes if they are clear and/or useful. They get downvotes when they are unclear, useless, and so on. This question might not seem "good" to you, but it is clear question, and it might be useful for some people to know the answer to it. I don't see a problem with upvoting it. –  Djaian Oct 27 '10 at 10:31

5 Answers 5

No. Weierstraß gave in 1872 the first published example of a contiuous function that's nowhere differentiable.

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36  
But using that example for this question is tantamount to using thermonuclear weapons to kill mosquitoes! –  Mariano Suárez-Alvarez Oct 26 '10 at 22:13
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As the last American president demonstrated, every potential threat deserves maximum military action. –  crasic Oct 26 '10 at 22:40

No, consider the example of $f(x) = |x|$. This function is continuous but not differentiable at $x = 0$.

There are even more bizare functions that are not differentiable everywhere, yet still continuous. This class of functions lead to the development of the study of fractals.

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For a nice simple example of an everywhere continuous, nowhere differentiable function it's hard to beat this example of John McCarthy.

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This is a nice example, and a nice exposition. It (the linked paper) also makes the point that (in some sense) most continuous functions are in fact non-differentiable. –  Matt E Oct 27 '10 at 4:58

The Wiener process is a continuous everwhere, but differentiable nowhere function (quite an impressive beast by the way...)

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More precisely, the sample paths are continuous and with probability $1$ nowhere-differentiable. –  Michael Greinecker May 15 '12 at 23:52

An interesting fact is that most (i.e. a co-meager set of) continuous functions are nowhere differentiable. The proof is a consequence of the Baire Category theorem and can be found (as an exercise) in Kechris' Classical Descriptive Set Theory or Royden's Real Analysis.

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This holds also for other notions of most. Also, a more accessible reference might be "Measure and Category" by Oxtoby. –  Michael Greinecker May 16 '12 at 0:09
    
@MichaelGreinecker Very interesting. Thanks. –  Quinn Culver May 16 '12 at 3:28

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