is always the non-negative real number the absolute value of a complex number is the square root which is therefore called the absolute square or squared modulus This generalizes the alternative definition for reals: The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity is a special case of multiplicativity that is often useful by itself. Absolute value function The real absolute value function is continuous everywhere. It is differentiable everywhere except for . It is monotonically decreasing on the interval and monotonically increasing on the interval . Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function. For both real and complex numbers the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself). Relationship to the sign function The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions: or and for , Derivative The real absolute value function has a derivative for every , but is not differentiable at . Its derivative for 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 at 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 with respect to 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. Antiderivative The antiderivative (indefinite integral) of the real absolute value function is where is an arbitrary constant of integration. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not. Distance The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for