Monotonically and strictly increasing functions

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.

• Yes. Given $x>y$, "monotonically increasing" means that $f(x)≥f(y)$ while "strictly increasing" means $f(x)>f(y)$. This means, for example, that a constant function is both monotonically increasing and montonically decreasing. – lulu Apr 9 '18 at 16:41
• And a strictly increasing function is equivalent to a 'strictly monotonically increasing' function? How do you define "strictly monotonically increasing"? I've never heard that. As lulu said, yes for everything else. – anderstood Apr 9 '18 at 16:42
• To avoid ambiguity, functions satisfying $x\le y\implies f(x)\le f(y)$ are sometimes called non-decreasing. – Julián Aguirre Apr 9 '18 at 16:42
• Thanks for your clarifications. @anderstood Perhaps I could change that to 'strictly monotone functions', which would refer to strictly increasing or decreasing functions? – T J. Kim Apr 9 '18 at 16:46
• I would avoid non-standard usage. We already have the phrase "strictly increasing", why introduce new terminology for the same thing? – lulu Apr 9 '18 at 16:48

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.

• I see, thanks for your answer. Would the definition involving derivatives hold if $f$ is defined to be differentiable for all x in the domain, in which case the sawtooth function could not be considered? – T J. Kim Apr 9 '18 at 16:58

A strictly increasing function: let $$P=\{x_1,...,x_n; x_i<x_{i+1}\}$$ then $f(x_i)<f(x_{i+1})$ for all $x_i,x_{i+1} \in P$. A monotonic increasing function: let $$P=\{x_1,...,x_n; x_i<x_{i+1}\}$$ then $f(x_i)\leq f(x_{i+1})$ for all $x_i,x_{i+1} \in P$. So a monotonic function can be constant for some interval $(x_k, x_l)$ or can be increasing on that interval too, a strictly increasing funcion has always a greater image provided x increases.