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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

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

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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
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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