Examples
- The absolute value function u : →, u(t) = |t|, which is not differentiable at t = 0, has a weak derivative v known as the sign function given by
- This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. Usually, this is not a problem, since in the theory of Lp spaces and Sobolev spaces, functions that are equal almost everywhere are identified.
- The characteristic function of the rational numbers is nowhere differentiable yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero,
- Thus is the weak derivative of . Note that this does agree with our intuition since when considered as a member of an Lp space, is identified with the zero function.
Read more about this topic: Weak Derivative
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