Suppose $f:\mathbb R \to \mathbb R$ is only differentiable at integer points. Is this possible? If does, what kind of function is $f$?
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If there are no other conditions on $f$, sure. – André Nicolas Jan 22 '15 at 03:31
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@AndréNicolas any examples..? I can't think of an example – user197137 Jan 22 '15 at 03:31
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2For example, $f(x) = 0$ if $x$ is irrational; and $f(x) = \sin^2(\pi x)$ if $x$ rational. – Simon S Jan 22 '15 at 03:34
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@SimonS: Easier to describe than mine, I was using $\frac{1}{q^2}$ for non-integer rationals $p/q$, $0$ elsewhere. – André Nicolas Jan 22 '15 at 03:37
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@AndréNicolas Thnks Alot~ – user197137 Jan 22 '15 at 03:39
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@SimonS Brilliant! Thnks Alot~ – user197137 Jan 22 '15 at 03:39
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@user197137: You are welcome. We can in the same way get a function differentiable precisely at the integers, and $k$ tmes differentiable there. With more effort, one can get infinitely ifferentiable. – André Nicolas Jan 22 '15 at 03:56