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
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3 Answers
62This 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.
62hmm.. on a second thought.. there is a slight(?) no big mistake in the above..
fidn it!
62These 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
