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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.

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    $\begingroup$ 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. $\endgroup$ – lulu Apr 9 '18 at 16:41
  • $\begingroup$ 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. $\endgroup$ – anderstood Apr 9 '18 at 16:42
  • $\begingroup$ To avoid ambiguity, functions satisfying $x\le y\implies f(x)\le f(y)$ are sometimes called non-decreasing. $\endgroup$ – Julián Aguirre Apr 9 '18 at 16:42
  • $\begingroup$ Thanks for your clarifications. @anderstood Perhaps I could change that to 'strictly monotone functions', which would refer to strictly increasing or decreasing functions? $\endgroup$ – T J. Kim Apr 9 '18 at 16:46
  • $\begingroup$ I would avoid non-standard usage. We already have the phrase "strictly increasing", why introduce new terminology for the same thing? $\endgroup$ – lulu Apr 9 '18 at 16:48
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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.

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  • $\begingroup$ 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? $\endgroup$ – T J. Kim Apr 9 '18 at 16:58
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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.

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