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Someone told me that a continuous function that is Lipschitz is differentiable almost every where. If this is true how do I see it?
Lipschitz Continuous Functions: Differentiable Almost Everywhere?
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A Lipschitz continuous function is indeed differentiable almost everywhere, which is supported by the Rademacher's theorem. The discussion highlights the connections between Lipschitz continuity, absolute continuity, and functions of bounded variation, noting that Lipschitz functions are absolutely continuous and thus have derivatives almost everywhere. It is emphasized that absolute continuity implies differentiability almost everywhere, while bounded variation does not guarantee this. The conversation also seeks relevant lemmas that affirm the differentiability of absolute continuous or bounded variation functions. Understanding these relationships is crucial for analyzing the properties of Lipschitz continuous functions.
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