| Monotonicity |
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In Mathematics , a monotonic function (or '''monotone function''') is a Function which 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 (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. 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. 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'')). The terms non-decreasing and non-increasing avoid any possible confusion with strictly increasing and strictly decreasing, respectively, see also Strict . Some basic applications and results In calculus, each of the following properties of a function ''f'' : R → R implies the next:
These properties are the reason why monotonic functions are useful in technical work in Analysis . Two facts about these functions are:
An important application of monotonic functions is in Probability Theory . If ''X'' is a Random Variable , its Cumulative Distribution Function FX is a monotonically increasing function. A function is '' Unimodal '' if it is monotonically increasing up to some point (the '' Mode '') and then monotonically decreasing. MONOTONICITY IN FUNCTIONAL ANALYSIS In Functional Analysis , an operator ''A'' from a Topological Vector Space ''V'' into its dual space ''V''∗ is said to be a monotone operator if |
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