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 calculus and analysis

In calculus, a function

f

{\displaystyle f}

defined on a subset of the real numbers with real values is called monotonic if it is either entirely non-decreasing, or entirely non-increasing. 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 termed monotonically increasing (also increasing or non-decreasing) if for all

x

{\displaystyle x}

and

y

{\displaystyle y}

such that

x

≤

y

{\displaystyle x\leq y}

one has

f

(

x

)

≤

f

(

y

)

{\displaystyle f\!\left(x\right)\leq f\!\left(y\right)}

, 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 x\leq y}

, then

f

(

x

)

≥

f

(

y

)

{\displaystyle f\!\left(x\right)\geq f\!\left(y\right)}

, 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 <}

, one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing). Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing).