6.4 Mean-Value Type Theorems 321- Prove that there is no differentiable function f, defined on any open
interval I containing 0, such that 't/x E I , f' ( x) = {
1
~f x ~ O, }.
21fx<O- Prove that if f is differentiable on an open interval I and 't/x EI, f'(x) #
0, then f'(x) has the same sign throughout the interval I. - Suppose f is differentiable on some neighborhood of x 0. Prove that if the
derivative f' has a discontinuity at x 0 , that discontinuity cannot be a
"removable" or "jump " discontinuity.^9 - Provocative Example:^1 ° Consider the function
f(x) = { x
2
+~x2
sin 1 ; 1 ifx#0, }·. 0 if x = 0.
(a) Prove that f has its absolute minimum at 0.
(b) Prove that f is differentiable everywhere, and find f' (0).
(c) Prove that Ve> 0, f'(x) has both positive and negative values in the
interval (O,c) and also in the interval (-c,O).
Thus, f has its absolute minimum value at 0 but its derivative does not
make a simple change of sign at O!
* 14. Cantor's Function: (for those that studied the Cantor set, Section 3.4,
and the Cantor function, Section 5.5). The Cantor function <p(x) : [O, 1] --+
[O, l] is monotone increasing on [O, 1], f(O) = 0, and f(l) = 1. Prove that
despite this monotonicity, <p^1 (x) = 0 everywhere on [O, 1] except possibly
on a nowhere dense set of measure 0. (In fact, <p does all its "rising" on
the Cantor set.)6.4 Mean-Value Type Theorems
You will recall (although perhaps dimly) two important theorems from ele-
mentary calculus: Rolle's theorem and the mean value theorem. You might
not remember why they are considered important. You will understand and
appreciate these theorems more after you have studied this section.
Theorem 6.4.1 (Rolle's Theorem) Let a< band suppose f: [a,b]--+ JR is
differentiable on (a,b) and continuous on [a,b], and f(a) = f(b). Then 3c E
(a, b) 3 f'(c) = 0.- See Definitions 5 .2.7 and 5.2.9.
- For another example, see page 36 of Gelbaum & Olmsted [49].