User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Theorem
Properties of Reciprocal Function:

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

Continuous

Differentiable

Smooth (inf. diffable)

Strictly decreasing on $\left(0\,.\,.\,+\infty\right)$

Strictly decreasing on $\left(-\infty\,.\,.\,0\right)$

Limits at infinity are zero


 * How do I prove smoothness? (Strong) induction? --GFauxPas 20:46, 2 August 2012 (UTC)


 * Mostly, it can be done through the 'composition of smooth functions is smooth' principle, but here I think it is required to write the $n$th derivative for each $n$ and prove it to be continuous. This may of course be done by induction, if you so desire. --Lord_Farin 20:49, 2 August 2012 (UTC)