Nondecreasing function ($x<y⟹f(x)\le f(y)$)

A real-valued function $f:D\to \mathbb{R}$ is a nondecreasing function if every pair $x<y$ in its domain of a function satisfies $f(x)\le f(y)$. Flat intervals are allowed, unlike for a strictly increasing function. For example,is a continuous nondecreasing function, but it repeats its output on each flat interval.

A nondecreasing function can also be discontinuous. The step function $F(x)=0$ for $x<0$ and $F(x)=1$ for $x\ge 0$ has a jump discontinuity at zero without ever decreasing. It is the cumulative distribution function of a random variable that equals zero with probability one; quantile functions handle such jumps through a generalized inverse. Nondecreasing allocation functions permit critical-value payments in single-parameter mechanisms.