- published: 12 Jan 2009
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In mathematics, a monotonic function (or monotone function) is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing or non-decreasing), if for all x and y such that x ≤ y one has f(x) ≤ f(y), so f preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever x ≤ y, then f(x) ≥ f(y), so it reverses the order (see Figure 2).
If the order ≤ 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(x) < f(y) or f(x) > f(y), thus f(x) is not equal to f(y)).
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