Maybe its a stupid question, sorry.
The question i ask my self is if every integral of a function, where the function is bounded and continuous on the interval of the integral than i can conclude that the function is integrable in that interval?
- The question is asked from the point of view of improper integrals.
Divide it to two:
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Intervals of the form $[a,b)$
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Intevals of the form $[a,\infty)$ – This is the part where i think i have a problem, see below.
*For intevals of the form $[a,b]$ its by definition integrable at that interval – for continuous functions
What i think:
For the first case: $[a,b)$ we can expand the interval to $[a,b]$, in that interval the function is continuous and bounded except, maybe, to finite number of points, therefore we can conclude its integrable.
For the second case and more problematic, $[a,\infty)$ namely: $\int_{a}^{\infty}f(x)dx$ can i prove it or is it even a false claim?
Thanks for the help.
Best Answer
In the case of intervals of the type $[a,b)$, you are right: extende $f$ to $[a,b]$ putting, say, $f(b)=0$. The this new function is bounded and it has a single point at which it is discontinuous, at most. So, yes, it will be integrable.
But for intervals of the type $[a,\infty)$ this doesn't work: the integral $\int_a^\infty1\,\mathrm dt$ doesn't converge.