Reciprocal Function is Strictly Decreasing

Theorem
The reciprocal function:


 * $\operatorname{recip}: \R \setminus \left\{ {0} \right\} \to \R$, $x \mapsto \dfrac 1 x$

is strictly decreasing:


 * on the open interval $\left ({0 \,.\,.\, +\infty} \right)$


 * on the open interval $\left ({-\infty \,.\,.\, 0} \right)$

Warning
Though the reciprocal function is decreasing on $\left ({-\infty \,.\,.\, 0} \right)$ and on $\left ({0 \,.\,.\, +\infty} \right)$, it is not decreasing on  $\left ({-\infty \,.\,.\, 0} \right) \cup \left ({0 \,.\,.\, +\infty} \right)$.

This is because there is a nonremovable discontinuity at the origin.

Also see

 * Reciprocal Sequence is Strictly Decreasing
 * Sum of Reciprocals is Divergent: Proof 2
 * Existence of Euler-Mascheroni Constant