Monotone function

A function between ordered sets is monotone if it preserves order: whenever one input is less than or equal to another, their images have the same relation.

Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Monotonicity is defined by comparing inputs and outputs in such ordered sets.
Monotonicity inequality: The condition x ≤ y implies f(x) ≤ f(y) for a monotone function. This implication is the defining test for order preservation.
Strictly increasing function: A function for which x < y always implies f(x) < f(y). Strict increase strengthens the weak inequality required for monotonicity.
Monotone convergence theorem: A theorem stating that an increasing sequence of nonnegative measurable functions has an integral limit equal to the limit of its integrals. Monotone behavior lets integrals pass to a limit under precise hypotheses.
Order relation: A binary relation that compares elements according to specified ordering properties. The function preserves this relation between each comparable pair of inputs.
Function composition: The operation of applying one function to the output of another. Composing monotone functions preserves monotonicity.
Antitone function: A function that reverses order, sending x ≤ y to f(x) ≥ f(y). It reverses the comparison that monotone functions preserve.
Monotone operator: An operator whose input-output relation satisfies a nonnegative inner-product condition. Operator theory extends order-like monotonicity to optimization and differential equations.
Nondecreasing function: A real-valued function whose value never decreases as its input increases. This is the usual real-variable form of order preservation.
Meet-semilattice: A partially ordered set in which every pair has a greatest lower bound. Monotone maps on these structures interact with their meet operations.

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