This is a question on terminology.
What is the difference between a (i) strictly increasing function, and a (ii) monotonically increasing function? Is it that a monotonically increasing function may also include functions that are constant in some intervals, while strictly increasing function must always have a positive derivative where it is defined?
If so, is it correct to say, that
Strictly increasing functions $\implies$ monotonically increasing, while the converse is not true? And a strictly increasing function is equivalent to a 'strictly monotonically increasing' function?
Thanks.
Best Answer
You almost have it right. The condition is better stated without referring to derivatives. A function $f(x)$ is strictly increasing if for all $(x,y)$ such that $y>x$,
$$ f(y) > f(x) $$
and is monotonic increasing if for all $(x,y)$ such that $y>x$, $$ f(y) \geq f(x) $$
Your definition involving derivatives would say that the sawtooth $$ g(x) = x - \lfloor x \rfloor $$ is strictly monotonic (since the derivative is not defined at integer $x$), but it is not monotonic at all.
Your last sentence is completely correct.