Homework 5 Solution

Description

Problem 21 (Exercise 5.15). Suppose f is defined in a neighborhood of x, and suppose f (x) exists. Show that

_lim f (x + h) + f (x − h) − 2f (x) _{= f (x).}

h→0 h²

Show by an example that the limit may exist even if f (x) does not exist.

Problem 22 (Exercise 6.5). Suppose f is a bounded real function on [a, b], and f ² ∈ R on [a, b]. Does it follow that f ∈ R? Does the answer change if we assume that f ³ ∈ R?

Problem 23 (Exercise 6.6). Let P be the Cantor set constructed in Sec. 2.44. Let f be a bounded real function on [0, 1] which is continuous at every point outside P . Prove that f ∈ R on [0, 1].

Problem 24 (Exercise 6.11). Let α be a fixed increasing function on [a, b]. For u ∈ R(α), define

1/2 b

u ₂ = |u|²dα .

a

Suppose f, g, h ∈ R(α), and prove the triangle inequality

f − h ₂ ≤ f − g ₂ + g − h ₂

as a consequence of the Schwarz inequality, as in the proof of Theorem 1.37.

Problem 25 (Exercise 6.15). Suppose f is a real, continuously differentiable function on [a, b], f (a) = f (b) = 0, and

b

5