I have some questions on the topic of improper integrals. I'm using Thomas' calculus 11th edition for reference, but I have a handful of other books providing me with the same information.(adsbygoogle = window.adsbygoogle || []).push({});

When they are defining improper integrals, they work with the hypothesis that f(x) must be continuous on [itex][a,\infty)[/tex].

From there they define

[itex]\int_a^{\infty} f(x)dx = \lim_{b \to \infty} \int_a^b f(x) dx. [/tex]

which is just fine. However, why do we need the hypothesis that f is continuous? I can't see any real need for this strong of a condition. I am thinking that that what they really want is boundedness on any finite subinterval of the real numbers, and forcing the function to be continuous is one way of doing this.

In the same way when they state the direct comparison test and the limit comparison test they require this condition of continuity. Is this for the same reason or a different reason?

Anyhow, I would greatly appreciate if there is anyone who could shine some light on this issue for me. It is a great frustration of mine that introductory calculus textbooks tend to refer to "more advanced texts" to justify and show certain results, yet those more advanced texts remain elusive to me.

Thanks,

-Xaenn

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# Improper Integrals

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