= Nondecreasing function
{title2=$x<y\implies f(x)\leq f(y)$}
= Nondecreasing
{synonym}
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 $f(x)\leq f(y)$. Flat <intervals> are allowed, unlike for a <strictly increasing function>. For example,
$$
p(x)=\min(1,\max(0,x))=\begin{cases}0,&x\leq0,\\x,&0<x<1,\\1,&x\geq1\end{cases}
$$
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 $x\geq0$ 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>.
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