When determining the intervals in which the graph of a function increase or decrease, some books include the ends while others do not.
Which one is true or which one is more preferable?
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$\begingroup$You can differentiate $f$ only in the open interval, that is, the interval without the endpoints. So, at first you can show that $f$ is increasing/decreasing in the interval without the ends.
But, if your function $f$ is defined on the endpoints and is continuous (as are probably most of the functions that you encounter) then it is correct to include also the endpoints.
Take for example the function $f(x)=\frac1x$. This is decreasing in, say $(0,1)$, but not in $[0,1)$ because is not defined in $x=0$. However we can correctly say that $f(x)=\frac1x$ is decreasing in $(0, 1]$ (in this endpoint, $x=1$ there is no problem, $f$ is continuous).
$\endgroup$ 3 $\begingroup$I assume the function is defined on the closed interval (i.e. $[a,b]$, endpoints included), if it weren't then you surely couldn't include the endpoints. According to the definition of an increasing function:
$$f(x):[a,b]\to\Bbb{R} \text{ is increasing } \iff f(x)<f(y)$$ $$\forall x,y\in[a,b] \text{ such that } x<y$$ So once you find out the function is increasing in the open interval $(a,b)$ by using differentiation criteria, then you can manually check that the conditions apply to the endpoints by showing that $$\lim_{x \to a^+} f(x)≥f(a) \text{ and } \lim_{x \to b^-} f(x)≤f(b)$$ Note that these conditions will be already given if the function you're dealing with is continuous in $a$ and $b$. In fact it can be easily proven that any continuous function defined on a closed interval and monotonic on the open interval with the same endpoints is also monotonic on the closed interval.
This shows that it isn't incorrect to exclude the endpoints, but it consists in a loss of information if the conditions are actually met.
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