Introduction
The following is a standard theorem from calculus.
Theorem
If the function [latex]f[/latex] is continuous on the closed interval [latex][a,b][/latex] and if [latex]f'(x)>0[/latex] in the
interior of the interval, then [latex]f[/latex] is increasing over [latex][a,b][/latex].
interior of the interval, then [latex]f[/latex] is increasing over [latex][a,b][/latex].
It is possible to replace the condition [latex]f'(x)>0[/latex] in the interior by the weaker
condition [latex]f'(x)[/latex] does not take the value 0 in the interior, and still draw the
conclusion that [latex]f[/latex] is strictly monotone over [latex][a,b][/latex].
It will also be shown that the interval need not include endpoints or may be infinite.
