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Old 2006-06-02, 17:29   #9
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Sep 2002
Rep├║blica de California

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Originally Posted by Damian
Has anybody one example of a function that is differentiable at a point, but is not a class C1 function at that point?
Is that even possible?
No - to begin with a function can in some sense never be "differentiable at a point", i.e. without referring to the function's values in a neighborhood of the point. A function is only differentiable at a point x0 if the limit of the divided difference [f(x)-f(x0)]/[x-x0] exists at that point, which implies that the limit exists and is the same, irrespective of the direction from which one approaches x0 in taking the limit. (I.e. from the left or the right if one restricts oneself to the real axis, or from an arbitrary direction if one generalizes to the complex plane.) For example, the function f(x) = |x| is continuous everywhere and the above limit has a well-defined value for a given direction of approach to x0 = 0 (-1 when approaching from the left, +1 from the right), but because the limit depends on the direction of approach the derivative does not in fact exist at x0 = 0. Since "C1" is equivalent to "derivative exists", if the function is not differentiable at a given point, it cannot be C1 there, and vice versa.

There are eminently readable online links to all of this stuff, you know - for example:
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