Assignment 4 of Math 5302
1. Show that
B(α, β) = Z 1
0
x
α−1
(1 − x)
β−1
dx
is well-defined for α > 0 and β > 0.
2. Show that if f is Riemann integrable on [a, b], then
lim
→0+
Z b−
a
f(x)dx =
Z b
a
f(x)dx.
3. Evaluate R 1
0
(1 − x
2
3 )
3
2 dx. Hint: Express the integral in terms of the gamma function first.
4. Let
f(x) =
x sin( 1
x
) if 0 < x ≤ 1;
0 if x = 0.
Show that f is bounded and continuous on [0, 1], but not of bounded variation on [0, 1].
5. Assume f is differentiable on [a, b] with |f
0
(x)| ≤ M < ∞ for a ≤ x ≤ b. Show that f is of
bounded variation and V
b
a
(f) ≤ M(b − a). (Hint: Use Mean Value Theorem.)
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