prove if $f$ is bounded in [a,b] and integrable in every [a+$\epsilon$ , b] for every $\epsilon$ > 0 then $f$ integrable in [a,b]

Question

I've started with showing that the improper integral in [a,b] is the limit when t goes to a+ (I've said that t=a+ $\epsilon$ ) but now to say that the improper integral is defined I need to show that the limit exist and final.

Answer 1

Let $M=\sup_{[a,b]}| f|$.

We can assume $M>0$.

Given $\epsilon>0$ small enough.

$f$ is integrable at $[a+\frac{\epsilon}{4M},b]$ thus by Cauchy criterion,

there exist a partition $\sigma$ of $[a+\frac{\epsilon}{4M},b]$ such that

$$U(f,\sigma)-L(f,\sigma)<\frac{\epsilon}{2}$$

Put $$\Sigma=\sigma \cup\{a\}$$

then

$$U(f,\Sigma)-L(f,\Sigma)\le$$ $$2M\frac{\epsilon}{4M}+U(f,\sigma)-L(f,\sigma)$$ $$<\epsilon$$

done.