Signpreserving

Signpreserving describes a property of a map, function, or operator that does not change the sign of a quantity. In real-valued terms, a function f: R -> R is sign-preserving if, for every nonzero input x, the sign of f(x) matches the sign of x. Concretely, this means x > 0 implies f(x) > 0 and x < 0 implies f(x) < 0. Some authors additionally require f(0) = 0 to maintain a coherent sign at zero.

Formal variants of the definition exist. The strongest common form requires that for all x ≠ 0, sign(f(x)) = sign(x). Under this interpretation, f is sign-preserving if it preserves the positive and negative halves of the real line. Weaker formulations may allow f(0) to take any value or may require preservation only on one half-line (positive or negative inputs).

Examples and non-examples help illustrate the concept. Functions of the form f(x) = c x with c > 0 are sign-preserving, as they scale but do not alter the sign of any nonzero input. The odd, strictly increasing functions with f(0) = 0, such as f(x) = x^3 or f(x) = x^5, are typically sign-preserving. In contrast, f(x) = x^2 or f(x) = e^x do not preserve sign for negative inputs, since negative x can produce nonnegative outputs.

Signpreserving properties are discussed in real analysis and numerical analysis, where preserving the sign can reflect physical constraints or stability considerations. They relate to, but are distinct from, monotone or order-preserving properties.