Tuesday, August 25, 2015

Non-differentiability implies non-Lipschitz continuity

In a simple one-dimensional framework, it is known that the differentiability of a function (with bounded derivative) on an interval implies its Lipschitz continuity on that interval. However, non-differentiability does not implies non-Lipschitz continuity as shown by the function f:x|x|. Still, there are functions that are not differentiable at a point and this argument is used to say that this same function is not Lipschitz-continuous at that point as, for example, f:xx at x=0 (we say that the slope of the function at that point is "vertical"). So my question is: is there a theorem telling us that for some functions (to be characterized), their non-differentiability implies their non-Lipschitz continuity? Or is this result obvious from the definition of Lipschitz continuity?

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