linguistic:translation
Absolute value function

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (−∞, 0 ] and monotonically increasing on the interval [ 0, +∞). 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).