Reciprocal Function is Strictly Decreasing

Theorem
The reciprocal function:


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

is strictly decreasing:


 * on the open interval $\openint 0 \to$


 * on the open interval $\openint \gets 0$

Warning
Though the reciprocal function is decreasing on $\openint \gets 0$ and on $\openint 0 \to$, it is not decreasing on $\openint \gets 0 \cup \openint 0 \to$.

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

Also see

 * Reciprocal Sequence is Strictly Decreasing
 * Harmonic Series is Divergent
 * Existence of Euler-Mascheroni Constant