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# Monotonicity and boundedness

A function $f$ is bounded if, for some constant $M$ and all $x\in D_f$, $$\vert f(x) \vert \le M.$$ It is bounded below if $f(x)\ge -M$ and bounded above if $f(x)\le M$.

The graph of a bounded function lies within two horizontal lines. The graph of a function bounded below remains above an horizontal line.

$\sin(x)$ is bounded; $x^2$ is bounded below; $x^3$ is unbounded.

A function is increasing if, whenever $x\gt y$, $$f(x)\ge f(y).$$ It is decreasing if $f(x)\le f(y)$. A function is strictly increasing or strictly decreasing if the inequalities are all strict. A function is monotonic if it is either always increasing or always decreasing.

The graph of an increasing function "climbs up or stays flat" as $x$ increases.

$x^3$ is strictly increasing; $e^{-x}$ is strictly decreasing; $\max(x,0)$ is increasing, but not strictly increasing when $x\lt 0$.