TITLE: Why is union sign not allowed to denote intervals of function increasing?
QUESTION [22 upvotes]: On the calculus exam, we were given a function $$f(x) = \frac{2x^2+x-1}{x^2-1}
$$ and asked to find intervals of function increasing and decreasing. In the post-exam solutions PDF, it was written that f(x) is increasing on $$(-\infty; -1), (-1; 1), (1; +\infty)$$
And specifically noted that writing the function increase interval as $$(-\infty; -1) \cup (-1; 1) \cup (1; +\infty)$$ is wrong and unacceptable, however, I fail to understand why so.
So, why is union-sign notation incorrect?

REPLY [2 votes]: The existing answers have focused on the fact that $f$ is not increasing from one interval to the next.  An even better example would be $g(x)$ defined as $-1/x$ when $x<0$ and $\sqrt{x}$ when $x\ge 0$.  Then $g$ is increasing on $(-\infty,0);[0,\infty)$.  But we definitely can't say that $g$ is increasing on $(-\infty,0)\cup[0,\infty)=(-\infty,\infty)$.