Editing Rolle's theorem (section)

== Generalization ==

The second example illustrates the following generalization of Rolle's theorem:

Consider a real-valued, continuous function {{mvar|f}} on a closed interval {{closed-closed|''a'', ''b''}} with {{math|1=''f&thinsp;''(''a'') = ''f&thinsp;''(''b'')}}. If for every {{mvar|x}} in the open interval {{open-open|''a'', ''b''}} the [[One-sided limit|right-hand limit]]
<math display="block">f'(x^+):=\lim_{h \to 0^+}\frac{f(x+h)-f(x)}{h}</math>
and the left-hand limit
<math display="block">f'(x^-):=\lim_{h \to 0^-}\frac{f(x+h)-f(x)}{h}</math><!-- The notation "lim as h tends to 0 minus" means h is negative. See example 2 at [[One-sided limit#Examples]] -->

exist in the [[extended real line]] {{closed-closed|−∞, ∞}}, then there is some number {{mvar|c}} in the open interval {{open-open|''a'', ''b''}} such that one of the two limits
<math display="block">f'(c^+)\quad\text{and}\quad f'(c^-)</math>
is {{math|≥ 0}} and the other one is {{math|≤ 0}} (in the extended real line). If the right- and left-hand limits agree for every {{mvar|x}}, then they agree in particular for {{mvar|c}}, hence the derivative of {{mvar|f}} exists at {{mvar|c}} and is equal to zero.

===Remarks===
* If {{mvar|f}} is convex or concave, then the right- and left-hand derivatives exist at every inner point, hence the above limits exist and are real numbers.
* This generalized version of the theorem is sufficient to prove [[Convex function|convexity]] when the one-sided derivatives are [[monotonically increasing]]:<ref>{{citation |last=Artin |first=Emil |author-link=Emil Artin |translator-first= Michael |translator-last= Butler |title=The Gamma Function |orig-year=1931 |year=1964 |publisher=[[Henry Holt and Company|Holt, Rinehart and Winston]] |pages=3–4}}.</ref> <math display="block">f'(x^-) \le f'(x^+) \le f'(y^-),\quad x < y.</math>