Monotonic function

A function that is either nondecreasing or nonincreasing throughout a specified domain or interval. A strictly monotonic function preserves or reverses order with strict inequalities.

Partially ordered set: A set equipped with a relation that orders some pairs while allowing incomparable elements. Monotonicity generalizes from ordered numbers to functions between partially ordered sets.
Derivative: A limit describing a function's instantaneous rate of change at a point. A derivative with a consistent sign on an interval implies monotonicity there.
Strictly monotonic function: A function that strictly preserves or reverses order: larger inputs always yield strictly larger or smaller outputs. It is the stronger version of monotonicity that rules out constant stretches.
Inverse function: A function that reverses the input-output mapping of a bijective function. A continuous strictly monotonic function on an interval has an inverse on its image.
Real number: A number on the continuous number line, including rational and irrational numbers. Real-valued monotonicity relies on the usual order of real inputs and outputs.
Mean value theorem: A theorem stating that a differentiable function's secant slope equals a derivative at an intermediate point. It converts a consistent derivative sign into an order relation between function values.
Nonmonotonic function: A function that neither preserves nor reverses input order throughout the domain under consideration. It changes direction or otherwise violates both monotonicity conditions on the interval.
Bisection method: A root-finding algorithm that repeatedly halves an interval whose endpoint values bracket a root. Monotonicity can ensure a bracket contains at most one root.
Interval (mathematics): A subset of the real line containing every number between any two of its elements. Monotonicity is often asserted on an interval, not across the function's entire domain.
Critical point (mathematics): A point in a function's domain where its derivative is zero or undefined. Critical points divide differentiable functions into intervals where monotonicity can be tested.

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