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All the integrals below are understood in the sense of the Lebesgue.

problem 1: Prove the following equality which we used in class without proof. Assume that f integrable over [ 3; 3]. Then

1945_integration.jpg

for any h ε (-3, 2).

problem 2: Assume that f is integrable over [0, 1]. Show that

712_integration.jpg

is di erentiable a.e on (0, 1).

problem 3: Assume that f is continuous on [0, 1]. Suppose that

1873_limit.jpg

Then, |f(x) - f(y)| ≤ M |x-y|

for any x, y ε [0, 1]

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