Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/190

 Equating (D) and (E), we get

which is the Theorem of Mean Value.

The student should draw curves (as the one in §105 to show that there may be more than one such point in the interval; and curves to illustrate, on the other hand, that the theorem may not be true if $$f(x)$$ becomes discontinuous for any value of x between a and b (Fig. a, §105), or if $$f'(x)$$ becomes discontinuous (Fig. b, §105).

Clearing (44) of fractions, we may also write the theorem in the form

Following the method of the last section, let R be defined by the equation