Overview
A monotonic (or monotone) function is a real-valued function on an ordered domain that moves in a single direction: it never reverses from increasing to decreasing or vice versa. Informally, on any interval where a function is monotonic it is either always increasing or always decreasing. Monotonicity is a basic concept in calculus and order theory and is used to identify simple structure in otherwise complicated functions.
Definitions and types
Two common senses of monotonicity are used:
- Nondecreasing (also called monotone increasing): for x1 < x2 we have f(x1) ≤ f(x2).
- Nonincreasing (also called monotone decreasing): for x1 < x2 we have f(x1) ≥ f(x2).
If the inequalities are strict for every pair x1 < x2 the function is called strictly increasing or decreasing. Constant functions are both nondecreasing and nonincreasing.
Calculus perspective
When a function is differentiable on an interval, the sign of its derivative gives information about monotonicity: if f'(x) >= 0 for all x in the interval, the function is nondecreasing there; if f'(x) <= 0 throughout, it is nonincreasing. This implication follows from the mean value theorem. The converse need not hold: monotone functions can fail to be differentiable or can have f' = 0 on large sets. In fact, a monotone function on an interval has a derivative almost everywhere (in the Lebesgue sense) but the derivative may vanish on sets of positive measure.
Key properties
- Monotone functions on an interval have bounded variation and can have only jump (discontinuous) points; such discontinuities are at most countable.
- A strictly monotone continuous function on an interval is one-to-one and therefore has a continuous inverse; the inverse preserves monotonicity (an increasing function has an increasing inverse).
- Compositions: composing two increasing functions yields an increasing function; composing an increasing with a decreasing function may reverse monotonicity.
Examples and counterexamples
Elementary examples include f(x)=x and the exponential function, which are strictly increasing on the real line. The quadratic function f(x)=x^2 is not monotone on R but is monotone on [0,∞) and (−∞,0]. Trigonometric functions such as sin x are not monotone on R but are monotone on suitably small intervals where their derivative does not change sign. Constant functions are monotone but not strictly so.
Contexts, uses and distinctions
In analysis, monotonicity is used to study limits, convergence, integrals and invertibility. In order theory and computer science the adjective "monotone" often describes order-preserving maps between partially ordered sets; see order-preserving maps. For geometric intuition about instantaneous change see gradient and slope and for the role of stationary points use stationary point. For technical discussion of derivatives and monotonicity consult derivative-related resources.