Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

Monotonicity in calculus and analysis

In calculus, a function f defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely increasing or decreasing. It is called monotonically increasing (also increasing or nondecreasing), if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or nonincreasing) if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), so it reverses the order (see Figure 2).

If the order \leq in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for x not equal to y, either x < y or x > y and so, by monotonicity, either f\!\left(x\right) < f\!\left(y\right) or f\!\left(x\right) > f\!\left(y\right), thus f\!\left(x\right) is not equal to f\!\left(y\right).)

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