There exist a sequence of continuous functions {$f_n$} on $\mathbb{R}$ such that {$f_n$} converges to $f$ uniformly on $\mathbb{R}$,but

$$\lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}f_n(x)dx \neq \int_{-\infty}^{\infty}f(x)dx$$

solution i tried-

as we know that if sequence of continous function $f_n$ converges to a fucntion $f$ over a given interval $[a,\infty)$ then $$\lim_{n\rightarrow\infty}\int_{a}^{\infty}f_n(x)dx = \int_{a }^{\infty}f(x)dx$$ so in the above question they replace the' $a$' with $-\infty$ does it effects the result?

Please help

Thank you


Your method is not correct. Actually, for this one, you find a counterexample for the existence.

Consider $$f_n(x)=\frac{1}{n(1+\vert x \vert)}$$ Now try to complete the details!

  • $\begingroup$ does it only work when interval is compact? $\endgroup$ – Honey Kumar Jun 12 at 15:41
  • $\begingroup$ The indefinite integral is divergent here $\endgroup$ – Chinnapparaj R Jun 12 at 15:47
  • 1
    $\begingroup$ Oh...you mean ..about original problem? Yes, for compact intervals , limit of the integral equals integral of the limit $\endgroup$ – Chinnapparaj R Jun 12 at 15:54

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