Homework 5 Solution

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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|2 .

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

f 2(x) dx = 1.

a

Prove that

b

a

xf (x)f (x) dx =2

1

and that

b

b

[f (x)]2 dx · a

a

x2f 2(x) dx > 4 .

1

5


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