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 h^{2}
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}dα .
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. |
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a |
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Prove that |
b |
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_{a} |
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xf (x)f (x) dx = − _{2} |
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1 |
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and that |
b |
b |
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[f (x)]^{2} dx · _{a} |
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a |
x^{2}f ^{2}(x) dx > _{4} . |
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1 |
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