Decreasing Monotone Function

analysis

Definition

Decreasing Monotone Function

A function $f:D\to \mathbb{R}$ is called decreasing monotone (or monotonically decreasing) on an interval $I\subseteq D$ if for all $x,y\in I$:

$$x<y⟹f(x)\ge f(y)$$

In this case, the function never increases as the input value grows.

Properties

• Relation to Derivative: If $f$ is differentiable on an open interval $I$, then $f$ is decreasing monotone if and only if its derivative is non-positive:$f^{\prime }(x)\le 0$ for all $x\in I$.
• Strict Monotonicity: If the inequality is strict ($f(x)>f(y)$ for $x<y$), the function is called strictly decreasing monotone.
• Limits: A bounded decreasing function always has a limit as $x\to \infty$.
• Algebra: The sum of two decreasing functions is decreasing.