# sequence of continous function

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

## 1 Answer

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!

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