TITLE: Difference between strictly increasing and increasing functions
QUESTION [6 upvotes]: What is the precise difference between strictly increasing and increasing functions??
I see these terms being thrown around a lot
My guess is that strictly increasing mean that derivative is only greater than 0 and in case of just increasing derivative can be greater than or equal or 0?

REPLY [3 votes]: Note that your inituition about derivatives is only $3$ quarters correct: If the function $f$ under consideration is differentiable in the considered interval open $I$, then
$$ f \text{ is increasing in } I  \Longleftrightarrow f'(x) \ge 0\; \forall x \in I.$$
OTOH, under the same conditions only one  of the implications holds for strictly increasing functions:
$$ f \text{ is strictly increasing in } I  \Longleftarrow f'(x) > 0\; \forall x \in I.$$
The other implication is wrong, for example $f(x)=x^3$ is strictly increasing in $I=(-1,1)$, while the derivative is zero at $x=0$.