Monotonic Function

A monotonic function is a function whose values consistently move in one direction as its inputs increase: they never decrease, or they never increase. For functions of a real variable, monotonicity describes an ordering property rather than smoothness or a constant rate of change. A monotonic function may have flat portions or jumps, whereas a strictly monotonic function must assign strictly ordered values to distinct ordered inputs. These distinctions are fundamental in mathematical analysis. (jirka.org)

Definitions and terminology

Let $f:D\to\mathbb R$, where the domain $D$ is a subset of $\mathbb R$. For every $x,y\in D$ with $x<y$, the four principal conditions are:

• Nondecreasing: $f(x)\le f(y)$.
• Nonincreasing: $f(x)\ge f(y)$.
• Strictly increasing: $f(x)<f(y)$.
• Strictly decreasing: $f(x)>f(y)$.

A function is monotonic if it is nondecreasing or nonincreasing, and strictly monotonic if it is strictly increasing or strictly decreasing. Terminology varies: some authors use “increasing” for nondecreasing, while others intend strict increase. Explicit inequalities remove this ambiguity. Constant functions satisfy both non-strict conditions but, on domains containing two or more points, neither strict condition. (jirka.org)

Monotonicity depends on the chosen domain. The polynomial $f(x)=x^2$ decreases strictly on $(-\infty,0]$ and increases strictly on $[0,\infty)$, but is not monotonic on all of $\mathbb R$. The function $f(x)=x^3$ is strictly increasing everywhere. These examples illustrate that the property concerns comparisons between every pair of inputs, not merely the appearance of part of a curve. (jirka.org)

Derivatives and monotonicity

For a differentiable function on an interval, the sign of its derivative characterizes non-strict monotonicity:

Similarly, $f'\le0$ characterizes nonincreasing functions. At included endpoints, the relevant continuity or one-sided differentiability conditions must also hold. The interval assumption matters because the proof compares two inputs using the mean value theorem:

for some $c\in(x,y)$. Thus a nonnegative derivative prevents a decrease. Conversely, the difference quotients of a differentiable nondecreasing function are nonnegative, so their limit is nonnegative. (jirka.org)

The stronger condition $f'(x)>0$ everywhere is sufficient for strict increase, but is not necessary. For $f(x)=x^3$, the derivative $3x^2$ vanishes at zero without destroying strict monotonicity. A derivative that vanishes everywhere on an interval, however, forces the function to be constant. These distinctions are useful in calculus when determining intervals of increase and decrease. (jirka.org)

Limits and discontinuities

A real-valued monotonic function on an interval has finite one-sided limits at every interior point. For a nondecreasing function,

It is a continuous function at $c$ precisely when these three quantities agree. Otherwise, its interior discontinuity is a jump discontinuity: the left and right limits differ. Monotonicity therefore excludes the oscillatory behavior possible for unrestricted functions near a discontinuity. (jirilebl.github.io)

The discontinuities form at most a countable set. To see why, assign a rational number inside each open interval $(f(c^-),f(c^+))$ associated with a jump. Distinct jumps have disjoint such intervals, so they receive distinct rationals. Nevertheless, a monotonic function can have discontinuities at every rational point of an interval; countability does not imply that jumps are isolated. At boundary points, one-sided limits may be infinite if the function is unbounded. (jirilebl.github.io)

Inverses

Strict monotonicity implies that a function is injective, so it has an inverse function on its image. The inverse of a strictly increasing function is strictly increasing; the inverse of a strictly decreasing function is strictly decreasing. If the original function is continuous on an interval, its image is an interval and its inverse is continuous there. (jirka.org)

For example, restricting $x^2$ to the nonnegative real numbers produces an invertible function whose inverse is the nonnegative square-root function. Conversely, a continuous injective real-valued function on an interval must be strictly monotonic. Neither injectivity alone nor continuity alone is enough: the interval and both properties are essential to this converse. (jirka.org)

Integration and probability

Every real-valued monotonic function on a closed bounded interval $[a,b]$ is Riemann integrable. For a nondecreasing function, an equal-width partition with $n$ subintervals gives upper and lower sums satisfying

This difference tends to zero, establishing integrability even when jumps are present. (jirka.org)

In probability, the cumulative distribution function of a random variable $X$,

is nondecreasing because the event $\{X\le x\}$ is contained in $\{X\le y\}$ whenever $x<y$. It is also right-continuous. Its jumps record point probabilities:

Thus the analytic structure of monotonic functions accommodates continuous distributions, discrete distributions, and distributions combining both kinds of behavior. (ericwb.me)