Monotone Function¶

Definition¶

Let $X\subseteq\mathbb{R}\text{.}$ A function $f\colon X\to \mathbb{R}$ is called (monotonically) increasing or nondecreasing if for all $x$ and $y$ in $X$ such that $x< y\text{,}$ one has $f(x)\leq f(y)\text{.}$ Likewise, a function is called (monotonically) decreasing or nonincreasing if whenever $x< y\text{,}$ then $f(x)\geq f(y)\text{.}$

A function $f\colon X\to \mathbb{R}$ is called strictly increasing if for all $x$ and $y$ in $X$ such that $x< y\text{,}$ one has $f(x)<f(y)\text{.}$ Likewise, a function is called strictly decreasing if whenever $x< y\text{,}$ then $f(x)> f(y)\text{.}$

Interpretation¶

A monotone function preserves or reverses order. If $f$ is increasing, a larger input never gives a smaller output; if $f$ is decreasing, a larger input never gives a larger output. Monotonicity rules out oscillation and simplifies qualitative analysis.

In the left panel the dashed lines compare $f(x)$ and $f(y)$ for $x<y$$y$, and indeed $f(x)\le f(y)$$f(y)$. The flat piece in the middle is allowed because the inequality in the definition is weak, but it is exactly what stops this function from being strictly increasing. The graph in the right panel only falls from left to right; it levels off for a moment near the middle but never stays flat, so the function is still strictly decreasing.

The definition compares any two inputs $x<y$ of $X$, not only nearby ones, so it is a statement about the whole set $X$. Terminology differs between texts: some use "increasing" for what this page calls strictly increasing, which makes "nondecreasing" the unambiguous word for $x<y\implies f(x)\le f(y)$$f(y)$. A constant function is both increasing and decreasing.

A common slip is to combine pieces. The function $1/x$ is decreasing on $(-\infty,0)$ and on $(0,+\infty)$, but not on its whole domain: $-1<1$ and yet $1/(-1)=-1<1=1/1$$1/1$.

A strictly monotone function is injective, so it has an inverse on its range; this is how logarithms invert exponential functions. Monotone sequences are the same notion for functions of the index $n$. In probability, every cumulative distribution function turns out to be nondecreasing.

Key Results¶

• Monotone functions are injective on intervals where they are strictly monotone.
• Monotone functions may have jump discontinuities.
• Monotonicity constrains the derivative. Let $I$ be an interval, let $x_0$ be an interior point of $I\text{,}$ and let $f$ be differentiable at $x_0\text{.}$ If $f$ is nondecreasing on $I\text{,}$ then $f'(x_0)\geq0\text{;}$ if nonincreasing, then $f'(x_0)\leq0\text{.}$
• Sign test for strict monotonicity. Let $f$ be continuous on an interval $I$ and differentiable on its interior. If $f'>0$ throughout the interior, then $f$ is strictly increasing on $I\text{;}$ if $f'<0\text{,}$ strictly decreasing; if $f'\equiv0\text{,}$ constant. This is proved from the mean value theorem.
• Characterization on an open interval. For differentiable $f$ on an open interval $I\text{,}$ $f$ is nondecreasing on $I$ if and only if $f'\geq0$ on $I\text{,}$ and nonincreasing if and only if $f'\leq0\text{.}$
• For strict monotonicity only one implication survives: $f'>0$ everywhere implies strictly increasing, but the converse fails, since $f(x)=x^3$ is strictly increasing on $\mathbb{R}$ with $f'(0)=0\text{.}$

Examples¶

• The function $f(x)=x^3$ is strictly increasing on $\mathbb{R}\text{.}$
• The signum function is monotone but not strictly monotone.

Prerequisites¶

• Function

Related Concepts¶

• Monotone Sequence
• Injective Function
• Mean Value Theorem
• Cumulative Distribution Function

Sources¶

• Math I: Calculus · Lectures 4 and 7

Aliases: monotonic function Keywords: increasing function, decreasing function Symbols: infinity, real numbers, subset

Last updated: 20 October 2026 03:02