Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses 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 {displaystyle f} defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called monotonically increasing (also increasing or non-decreasing), if for all x {displaystyle x} and y {displaystyle y} such that x ≤ y {displaystyle xleq y} one has f ( x ) ≤ f ( y ) {displaystyle f!left(x ight)leq f!left(y ight)} , so f {displaystyle f} preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing) if, whenever x ≤ y {displaystyle xleq y} , then f ( x ) ≥ f ( y ) {displaystyle f!left(x ight)geq f!left(y ight)} , so it reverses the order (see Figure 2). If the order ≤ {displaystyle leq } in the definition of monotonicity is replaced by the strict order < {displaystyle <} , 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 {displaystyle x} not equal to y {displaystyle y} , either x < y {displaystyle x<y} or x > y {displaystyle x>y} and so, by monotonicity, either f ( x ) < f ( y ) {displaystyle f!left(x ight)<f!left(y ight)} or f ( x ) > f ( y ) {displaystyle f!left(x ight)>f!left(y ight)} , thus f ( x ) {displaystyle f!left(x ight)} is not equal to f ( y ) {displaystyle f!left(y ight)} .) If it is not clear that 'increasing' and 'decreasing' are taken to include the possibility of repeating the same value at successive arguments, one may use the terms weakly increasing and weakly decreasing to stress this possibility.