= Symmetric derivative criterion for monotonicity
{title2=$f^s>0\Longrightarrow f\text{ strictly increasing}$}
A <continuous function> on an open <interval> with positive <symmetric derivative> everywhere is <strictly increasing>. One proof takes an <antiderivative> $F$ using the <fundamental theorem of calculus>. For small positive $h$, the identity
$$
F(x+h)-2F(x)+F(x-h)=\int_0^h(f(x+t)-f(x-t))\,dt>0
$$
follows from $f^s(x)>0$. If $F$ exceeded a chord, its difference from the chord would attain a positive interior <maximum>, contradicting this symmetric difference <inequality>. Thus $F$ is a <convex function>, and its <derivative> $f$ is <nondecreasing>. Equality at two different points would make $f$ constant between them and force an interior <symmetric derivative> to vanish, proving strict <monotonicity>.
Back to article page
