Increasing and Decreasing Functions
A function is increasing where its graph rises from left to right and decreasing where it falls, determined by the sign of the first derivative.
Formula
f'(x) > 0: \text{ increasing; } f'(x) < 0: \text{ decreasing}
Definition
A function is increasing on an interval if the output goes up as you move right, and decreasing if the output goes down. Formally, $f$ is (strictly) increasing on $[a, b]$ if $a < b$ implies $f(a) < f(b)$; if $f$ is differentiable, $f' > 0$ on $(a, b)$ implies $f$ is increasing there, while $f' < 0$ implies decreasing, and critical points separate increasing and decreasing intervals. Monotone functions (non-decreasing everywhere) have at most countably many discontinuities and are measurable.
Example
$f(x) = x^2$ is decreasing for $x < 0$ (going downhill toward $0$) and increasing for $x > 0$ (going uphill away from $0$). For $f(x) = x^3 - 3x$: $f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$ is positive on $(-\infty,-1)$ and $(1,\infty)$ (increasing) and negative on $(-1,1)$ (decreasing). The inverse function theorem uses this idea: if $f$ is continuously differentiable with $f'(a) \neq 0$, then $f$ is locally strictly monotone near $a$, guaranteeing a local inverse function.
Key Insight
The sign of the slope (derivative) tells you the direction, positive slope means going up, negative slope means going down, and identifying increasing and decreasing intervals is a key step in graphing functions and solving optimization problems. Monotone functions are central to real analysis (they are Riemann integrable), measure theory (distribution functions of measures), and order theory (monotone mappings between posets).