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I was given two definitions:

  1. Let the function $f(x)$ be defined in a interval $[a,\infty)$ we will say that $f$ is locally integrable in $[a,\infty)$ if for all $a<b$ $f$ is integrable in $[a,b]$

  2. Let $f$ be defined and locally integrable in $[a,\infty)$ we will define the improper integral $\int_{a}^{\infty}f(x)dx$ to be $\lim\limits_{R \to \infty} \int_{a}^{R}f(x)dx$.
    a.if the limit exist and is finite we say that $\int_{a}^{\infty}f(x)dx$ converges and $f$ is integrable in $[a,\infty)$
    b.if the limit does not exist we say that $\int_{a}^{\infty}f(x)dx$ diverges and $f$ is not integrable in $[a,\infty)$

Is the definition 2 part b correct?

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  • $\begingroup$ What exactly is your question here? And 2)b looks right to me $\endgroup$ – kcborys Jul 26 '16 at 13:52
  • $\begingroup$ @Captain edited the question, yes I want to know if the definition is correct $\endgroup$ – gbox Jul 26 '16 at 13:54
  • $\begingroup$ It's unusual to say $f$ is integrable if that limit exists. $\endgroup$ – zhw. Jul 26 '16 at 22:57
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The second part looks right. It's actually the first one that could get some improvement. I would say:

Let the function $f(x)$ be defined on the interval $[a, +\infty)$. We say that $f$ is locally integrable if $f$ is integrable on every compact subset $[c, d] \subset [a, +\infty)$.

I think that your definition is not very clear, having multiple occurrences of $a$, but for different things.

If your confusion stems from the fact that the limit could be infinite, then know that some texts say that the integral diverges, other would write $\int_a^{+\infty} f(x)\,\mathrm dx = \infty$.

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