7.6 The Fundamental Theorem of Calculus 421- Let F(x) =fox T, where T denotes Thomae's function. Prove that
(a) Fis differentiable everywhere on [O, l].
(b) for all x in a dense subset of [O, 1], F'(x) =f. T(x).- Prove that \:/a < b, 3 f:[a, b] --t IR such that f is integrable on [a, b]
but there is no nonempty subinterval [c, d] ~ [a, b] on which f has an
antiderivative. [Hint: Use the function given in Theorem 5.7.3. Show that
this function has jump discontinuities on a dense subset of [a, b] and apply
Exercise 6.3.12.]
*20. Monotone f Implies One-Sided Differentiability of f~f: Suppose
f is nonnegative and monotone increasing on [a, b]. Then f is integrable
on [a, b] and the function F(x) = J: f is monotone increasing on [a, b].
(Justify.) Hence, at any point xo E (a, b), the four one-sided limitslim f(x), lim f(x), lim F(x), lim F(x)
x-+x 0 x-+x;j x-+x 0 x-+x;j
exist (see Theorem 5.2.17). Prove that f is differentiable from the left at
x 0 , and differentiable from the right at x 0 , andF!_(xo) = lim f(x), and F~(xo) = lim f(x).
x-+x 0 x-+x;j
(These one-sided derivatives were defined in Definition 6.1.11.) [Hint:
Revise the proof of Theorem 7.6.8.]*21. Application of the Previous Problem:
(a) Suppose f is nonnegative and monotone increasing on [a, b], and
x 0 E (a, b). Prove that if f is not continuous at x 0 , then the function
F(x) = J~"' f is not differentiable at xo.
(b) In Theorem 5.7.3 we proved that there is a bounded, monotone
increasing function f:[a, b] --t IR that is continuous at every irrational
number and discontinuous at every rational number in [a, b]. (See also
Exercise 7.4.19.) Use this and the result of (a) above to prove that there
exists a continuous, monotone increasing function that is differentiable at
every irrational number in [a, b] and nondifferentiable at every rational
number in [a, b].*22. Prove Theorem 7.6.20.
*23. Prove Equation (30) of Theorem 7.6.21.
*24. Prove Equations (32), (33), and (35) of Theorem 7.6.22.