Monotone function

A function is increasing when $x\le y$ implies $f(x)\le f(y)$.

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 nondecreasing function never decreases when its argument increases. Flat intervals are allowed, unlike for a strictly increasing function. 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 function is strictly increasing when $x<y$ implies $f(x)<f(y)$. It is therefore injective and each value has at most one preimage.

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.

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