Monotonic function

Intuition

While function congruence ($\forall f,a=b⟹fa=fb$) is regularly applied with equality, it does not hold universally for other relations.

Though it may not hold for all $f$, there are some functions for which congruence is applicable to inequalities. If a function $f$ satisfies $a\le b⟹fa\le fb$, it is called monotonically-increasing; or, for a strict order $<$, strictly-increasing.

Whilst a monotonically-increasing function preserves order and never decreases, stripping the qualifier "increasing" leaves us with a monotonic function — one that either never decreases, or never increases. That is, a monotonic function either preserves the given order, or reverses it. The former case is termed monotonically-increasing, and the latter is monotonically-decreasing.

Most relevantly to my college algebra class are the (partial) monotonicities of addition and multiplication on $ℝ$.

• $fx=x+c$ is monotonic for any $c\in ℝ$.

• $fx=cx$ is monotonic for non-zero $c$.

  • For $c>0$, $f$ is monotonically-increasing: $a<b⟹ca<cb$.

  • For $c<0$, $f$ is monotonically-decreasing: $a<b⟹ca>cb$.

  N.B. Within my college algebra course, what they call an "increasing function" is strictly-increasing function. Likewise for "decreasing function".