Monotonic function

A real-valued function $f:D\to \mathbb{R}$, with domain of a function $D\subseteq \mathbb{R}$ in the usual order of the real numbers, is a monotonic function if it is either a nondecreasing function throughout $D$ or a nonincreasing function throughout $D$. The same direction must work for every pair $x<y$ in $D$. Being a strictly increasing function on one part and a strictly decreasing function on another does not make the whole function monotone: for example, $x^2$ on the real line changes direction at zero. A constant function is both a nondecreasing function and a nonincreasing function.

The word "increasing" is ambiguous across textbooks: some allow equal outputs and some require unequal outputs. Use nondecreasing function for the weak inequality and strictly increasing function for the strict inequality; use nonincreasing function and strictly decreasing function in the other direction. Neither continuity nor differentiability is part of the definition of a monotonic function.

Strict monotonicity gives injectivity, so an equation $f(x)=b$ has at most one solution and an inverse function exists on the image of a function. Weak monotonicity permits repeated outputs and jumps. In probability theory, a cumulative distribution function is a nondecreasing function; its quantile function uses a generalized inverse to handle those flat parts and jumps. Monotonicity alone does not guarantee continuity, even for a strictly increasing function.

Let $I$ be a real interval, and let $f:I\to \mathbb{R}$ be differentiable in its interior and continuous at any endpoints belonging to $I$. If the derivative satisfies $f^{\prime }(x)\ge 0$ throughout the interior, then $f$ is a nondecreasing function on $I$. For $x<y$ in $I$, the mean value theorem supplies $c\in (x,y)$ withChanging $\ge 0$ to $\le 0$, $>0$, or $<0$ gives a nonincreasing function, strictly increasing function, or strictly decreasing function, respectively. The endpoint continuity assumption lets the mean value theorem apply even when $x$ or $y$ is an endpoint. ^[ref].

Conversely, if a nondecreasing function on a real interval is differentiable at an interior point $x$, then $f^{\prime }(x)\ge 0$: the difference quotient $(f(x+h)-f(x))/h$ is nonnegative for either sign of $h$. A differentiable nonincreasing function has derivative $f^{\prime }(x)\le 0$. Strict monotonicity still only forces these weak signs: the strictly increasing function $x^3$ has derivative zero at zero. The sign of the derivative helps locate intervals of monotonicity when sketching a graph. ^[ref].

The real interval hypothesis matters because a connected subset of the real line contains every point between its inputs. On a disconnected domain of a function, positive derivatives need not give a nondecreasing function. For example, takeHere $f^{\prime }(x)=1$ everywhere in $D$, but $-3/2<3/2$ while $f(-3/2)=-3/2>-17/2=f(3/2)$. Each component has a strictly increasing function, yet their outputs are ordered the wrong way across the gap. This function is neither a nondecreasing function nor a nonincreasing function on all of $D$; the mean value theorem cannot cross the missing interval.

A continuous function on an open interval with positive symmetric derivative everywhere is strictly increasing. One proof takes an antiderivative $F$ using the fundamental theorem of calculus. For small positive $h$, the identityfollows from $f^s(x)>0$. If $F$ exceeded a chord, its difference from the chord would attain a positive interior maximum, contradicting this symmetric difference inequality. Thus $F$ is a convex function, and its derivative $f$ is nondecreasing. Equality at two different points would make $f$ constant between them and force an interior symmetric derivative to vanish, proving strict monotonicity.

For a continuous strictly decreasing real function, its average $h(x)=x^{-1}{\int }_0^{x}f(t)dt$ lies strictly between $f(x)$ and $f(0)$ for $x>0$. There is a unique $g(x)\in (0,x)$ with $f(g(x))=h(x)$. If $f$ is differentiable and $f^{\prime }<0$ throughout the positive half-line, the fundamental theorem of calculus and the derivative formula for the monotone inverse giveContinuity of $f^{\prime }$ is unnecessary: differentiability of a strictly monotone function with nonzero derivative at a point suffices for differentiability of its inverse there.

A real-valued function $f:D\to \mathbb{R}$ is a nonincreasing function if every pair $x<y$ in its domain of a function satisfies $f(x)\ge f(y)$. Equal outputs and flat intervals are allowed. For example, $f(x)=-\min (1,\max (0,x))$ is a nonincreasing function, with a falling segment between two flat portions. Requiring $f(x)>f(y)$ instead gives a strictly decreasing function. Negating a nondecreasing function gives a nonincreasing function.

A nonincreasing sequence of natural numbers has finitely many strict drops, so it is eventually constant. This differs from a strictly decreasing sequence, which cannot continue indefinitely within the natural numbers.

// Target: foundations-of-mathematics.bigb

For a continuous unbounded nondecreasing function $f$ on the nonnegative half-line and $a>f(0)$, the non-strict inverse $h$ is left-continuous in its level argument. It equals the left limit of the strict generalized inverse of a nondecreasing function $G(a)=\inf \{t:f(t)>a\}$. Indeed $G(b)\le h(a)$ for $b<a$, while for every $t<h(a)$, $f(t)<a$ and choosing $b\in (f(t),a)$ gives $G(b)\ge t$. Similarly $h(a+)=G(a)$. A flat part of $f$ at level $a$ gives a discrepancy between the two inverses: $h(a)$ selects the beginning of the flat part and $G(a)$ its end. For a Brownian running maximum, these are the non-strict and strict passage times.

For positive summable weights $w_j$ at points $q_j$, the strict cumulative function $F(x)={\sum }_{q_j<x}{w}_j$ is nondecreasing and left-continuous. A finite-head/tail argument proves left-continuity, and its right jump at an atom is the mass there. This strict convention differs at atoms from the usual right-continuous cumulative distribution function.

A real-valued function $f:D\to \mathbb{R}$ is a nondecreasing function if every pair $x<y$ in its domain of a function satisfies $f(x)\le f(y)$. Flat intervals are allowed, unlike for a strictly increasing function. For example,is a continuous nondecreasing function, but it repeats its output on each flat interval.

A nondecreasing function can also be discontinuous. The step function $F(x)=0$ for $x<0$ and $F(x)=1$ for $x\ge 0$ has a jump discontinuity at zero without ever decreasing. It is the cumulative distribution function of a random variable that equals zero with probability one; quantile functions handle such jumps through a generalized inverse. Nondecreasing allocation functions permit critical-value payments in single-parameter mechanisms.

For an unbounded nondecreasing function $f:[0,\infty )\to \mathbb{R}$, its strict generalized inverse is $G(a)=\inf \{t\ge 0:f(t)>a\}$. It is nondecreasing and right-continuous in $a$. If $a_n\downarrow a$, then for any $t>G(a)$ monotonicity gives $f(t)>a$, so eventually $f(t)>a_n$ and $G(a_n)\le t$; also $G(a_n)\ge G(a)$. Local finiteness gives finite left limits. Flat parts of $f$ create jumps of $G$. For the continuous running maximum of Brownian motion, this gives the càdlàg Brownian first-passage subordinator.

A real-valued function $f:D\to \mathbb{R}$ is a strictly increasing function if every pair $x<y$ in its domain of a function satisfies $f(x)<f(y)$. It is therefore injective, and each output has at most one preimage. Its inverse function on its image of a function is also strictly increasing. Every strictly increasing function is a nondecreasing function; the converse fails when outputs repeat.

The polynomial $f(x)=x^3$ is a strictly increasing function on the real line, although its derivative $f^{\prime }(x)=3x^2$ is zero at zero. Indeed, for $x<y$,because $x^2+xy+y^2=(x+y/2{)}^2+3y^2/4$ can vanish only when $x=y=0$. Thus a positive derivative everywhere is sufficient on a real interval, but is not necessary for a strictly increasing function.

Strict monotonicity does not ensure continuity: $g(x)=x$ for $x<0$ and $g(x)=x+1$ for $x\ge 0$ is a strictly increasing function with a jump discontinuity. Its image of a function omits $[0,1)$, so the inverse function exists only on the actual image of a function, rather than on all the real numbers.

A strictly increasing function from a closed real interval onto the interval between its endpoint values is continuous. Given a desired output tolerance, choose nearby output values on either side and use their preimages to trap nearby inputs. At the endpoints only one side is needed. Equivalently, a discontinuity of a monotone function creates a gap in its range, which is incompatible with surjectivity onto an interval.

A real-valued function $f:D\to \mathbb{R}$ is a strictly decreasing function if every pair $x<y$ in its domain of a function satisfies $f(x)>f(y)$. It is a nonincreasing function and is injective, with a strictly decreasing inverse function on its image of a function. Negating a strictly increasing function gives a strictly decreasing function: $f(x)=-x^3$ is an example on the real line, even though its derivative vanishes at zero. Equal outputs are forbidden, unlike for a general nonincreasing function.

Every real-valued monotone function on a real interval is differentiable with a finite derivative at almost every point.

This strengthens the regularity known for a monotonic function beyond the restriction on its jump discontinuities.