calculus; rather analysis

suppose that f(x) is properly integrable over [a,b] and g(x) properly integrable over [a,b+d], d>0. then prove that

\lim_{\delta \rightarrow +0}\int_{a}^{b}{}f(x)g(x+\delta )dx=\int_{a}^{b}{f(x)g(x)}dx

3 Answers

62
Lokesh Verma ·

This is simple

let M be the maxima, m be the minima of f(x) over the interval [a,b]

now take \int_{a}^{b}{f(x)\left(g(x+\delta)-g(x) \right)}dx

m\int_{a}^{b}{\left(g(x+\delta)-g(x) \right)}dx\leq \int_{a}^{b}{f(x)\left(g(x+\delta)-g(x) \right)}dx\leq M\int_{a}^{b}{\left(g(x+\delta)-g(x) \right)}dx

take limit on both sides, you get that both are zero..

so by sandwich theorem you have proved the given question.

62
Lokesh Verma ·

hmm.. on a second thought.. there is a slight(?) no big mistake in the above..

fidn it!

62
Lokesh Verma ·

These are all higher mathematics problems.. so guys dont band your head with these..

they are from pure mathematics..

i think this question is from walter rudin ... if i remember correctly

btw I din understand what is properly integrable..

b555 do you mean Reimann integrable or Lebesgue Integrable

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