Absabsx

Absabsx is the composition of the absolute value function with itself, defined as f(x) = | | x | |. In real numbers, this simplifies to f(x) = |x| for all x, because the inner absolute value already yields a nonnegative quantity. Therefore absabsx is identical to the standard absolute value function on the real line.

Properties: f is an even function, since f(-x) = f(x); it is nonnegative, and it is piecewise linear with f(x) = x for x ≥ 0 and f(x) = -x for x ≤ 0. The graph has a cusp at x = 0, and the derivative is 1 for x > 0 and -1 for x < 0; the derivative at 0 is undefined in the classical sense, though subgradients include all values between -1 and 1.

Complex numbers: If z is a complex number, the modulus |z| is nonnegative and satisfies | |z| | = |z|. Consequently absabsx(z) = |z| as well, so nesting the modulus on complex inputs yields no new effect beyond the ordinary magnitude.

Programming perspective: In many languages, abs is a function returning a nonnegative value. Applying abs twice is redundant: abs(abs(x)) equals abs(x) for all commonly used numeric types, and in practice the inner call can be omitted.

Examples: absabsx(3) = 3; absabsx(-7) = 7; absabsx(0) = 0.