Absolute value
Derivative

The real absolute value function has a derivative for every x ≠ 0, but is not differentiable at x = 0. Its derivative for x ≠ 0 is given by the step function:

The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist.

The subdifferential of | x | at x = 0 is the interval.

The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.

The second derivative of | x | with respect to x is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the Dirac delta function.
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