I'm working on a problem that involves finding the intervals where a function $f$ is increasing and decreasing. Given the function$$ f(x) =
\cases{
x+7 & \text{if } x\lt -3\cr
|x+1| & \text{if } -3\le x <1\cr
5-2x & \text{if } x\ge 1\cr
}$$
I worked out that $f$ is increasing on $(-\infty,-3)$ and $[-1,1)$ , and $f$ is decreasing on $[-3,-1], [1,\infty)$.
However, the solution in the book claims that $f$ is increasing on $[-1,1]$, rather than $[-1,1)$ like I worked out. I am having trouble understanding why the book claims that this is the case.
I was under the impression that $f$ must be differentiable on the interior of an interval $I$ and continuous on all of $I$ in order to make any statements about increasing/decreasing behavior on the closed interval $I$. I'm not sure if I am overlooking something, but it seems that $f$ is not continuous on $[-1,1]$. I would appreciate any clarification.
Best Answer
It is true that if you have a differentiable function on an interval, then it is increasing if and only if its derivative is non-negative. However, increasing functions need not be differentiable according to their definition: $\def\rr{\mathbb{R}}$
A function $f : \rr \to \rr$ is increasing on a collection $S$ if and only if:
For any $x,y \in S$ such that $x \le y$:
$f(x) \le f(y)$.
Note that this definition is incompatible with the one that lulu proposed in a comment, but I believe this is the typical definition.