Basic Analysis I — Real Analysis · Problem 5.3.12

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · fundamental-theorem-of-calculus

Problem

(uses §3.6) a) Suppose

f : [a,b] \to \mathbb{R}

is increasing; by Proposition 5.2.11,

f

is Riemann integrable. Suppose

f

has a discontinuity at

c \in (a,b)

; show that

F(x) := \int_a^x f

is not differentiable at

c

. b) In Exercise 3.6.11, you constructed an increasing function

f : [0,1] \to \mathbb{R}

that is discontinuous at every

x \in [0,1] \cap \mathbb{Q}

. Use this

f

to construct a function

F(x)

that is continuous on

[0,1]

, but not differentiable at every

x \in [0,1] \cap \mathbb{Q}