Question

Monotonic increasing function vs increasing function

Answer 1



Figure 1. A monotonically increasing function.



Figure 2. A monotonically decreasing function



Figure 3. A function that is not monotonic

In mathematics, a monotonic function[1] [2](or monotone function[3]) 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.

Monotonicity in calculus and analysis

In calculus, a function

{\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. [2]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[3]), if for all

{\displaystyle x}

and

{\displaystyle y}

such that

{\displaystyle x\leq y}

one has

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

, so

{\displaystyle f}

preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing(also decreasing or non-increasing[3]) if, whenever

{\displaystyle x\leq y}

, then

{\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 <}

, then one obtains a stronger requirement. A function with this property is called strictly increasing[3]. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing[3]. Functions that are strictly increasing or decreasing are one-to-one(because for

{\displaystyle x}

not equal to

{\displaystyle y}

, either

{\displaystyle x<y}

or

{\displaystyle x>y}

and so, by monotonicity, either

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

or

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

, thus

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

is not equal to

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

.)

When functions between discrete sets are considered in combinatorics, it is not always obvious that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, so one finds the terms weakly increasing and weakly decreasing to stress this possibility.

The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing.

A function

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

is said to be absolutely monotonic over an interval

{\displaystyle \left(a,b\right)}

if the derivatives of all orders of

{\displaystyle f}

are nonnegative or all nonpositive at all points on the interval.

Monotonic transformation

The term monotonic transformation (or monotone transformation) can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences).[4]In this context, what we are calling a "monotonic transformation" is, more accurately, called a "positive monotonic transformation", in order to distinguish it from a “negative monotonic transformation,” which reverses the order of the numbers.[5]

Some basic applications and results

The following properties are true for a monotonic function

{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }

:

{\displaystyle f}

has limits from the right and from the left at every point of its domain;

{\displaystyle f}

has a limit at positive or negative infinity ( 

{\displaystyle \pm \infty }

 ) of either a real number,

{\displaystyle \infty }

, or

{\displaystyle \left(-\infty \right)}

.

{\displaystyle f}

can only have jump discontinuities;

{\displaystyle f}

can only have countably many discontinuities in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval (a,b).

These properties are the reason why monotonic functions are useful in technical work in analysis. Two facts about these functions are: