TITLE: Is $1/x$ strictly decreasing?
QUESTION [5 upvotes]: Is $1/x$ a strictly decreasing function over all $\mathbb{R}$? The question arises because when we differentiate$1/x$ we get $-1/x^2$ (all calculations exclude $0$), which is always negative and thus indicates strictly decreasing. But if we look at the graph then we see an anomaly at $0$. So, what should be the correct answer?

REPLY [3 votes]: We can define: $f$ is strictly decreasng on a set $A$, where $A$ is a subset of the domain of $f$.  In your case, $1/x$ is strictly decreasing on $(0,+\infty)$ or any subset of it; and $1/x$
is strictly decreasing on $(-\infty,0)$ or any subset of that.
But $1/x$ is not strictly decreasing on the set $\{-2,7\}$, for example.
The criterion with $f'(x) < 0$ can be used to prove $f$ is strictly decreasing on an interval.  But it cannot be used in the $\{-2,7\}$ example.