Bourbaki

Exercise 7, § 3

Functions of a Real Variable, II, 87

Let ff be a real function defined and having a continuous derivative on the interval ]0,+∞[]0, +\infty[, and such that

lim⁡x→0f(x)=lim⁡x→+∞f(x)=0. \lim_{x \to 0} f(x) = \lim_{x \to +\infty} f(x) = 0.

The integral ∫0+∞f′(αt)dt\int_0^{+\infty} f'(\alpha t) dt is defined and continuous on every bounded interval ]0,a]0, a]: show that the integral ∫0+∞dt∫0af′(αt)dα\int_0^{+\infty} dt \int_0^a f'(\alpha t) d\alpha may either not exist or may be different from

∫0adα∫0+∞f′(αt)dt. \int_0^a d\alpha \int_0^{+\infty} f'(\alpha t) dt.

Solution

No solution has been written yet.

Transcribed from the printed volume. Not translated.