- #1

- 732

- 188

## Summary:

- Let ##f:[a,b] \mapsto \mathbb R## be a function a partition ##P= \{x_0, x_2 \cdots , x_n \}## of ##[a,b]## is there such that, ##g:[a,b] \mapsto \mathbb R## , ##g(x) = f(x) ~~~x\in(x_{i-1}, x_i)##. Given that ##g## is integrable prove that ##f## is integrable.

## Main Question or Discussion Point

We have a function ##f: [a,b] \mapsto \mathbb R## (correct me if I'm wrong but the range ##\mathbb R## implies that ##f## is bounded). We have a partition ##P= \{x_0, x_1 , x_2 \cdots x_n \}## such that for any open interval ##(x_{i-1}, x_i)## we have

$$

f(x) =g(x)

$$

(##g:[a,b] \mapsto \mathbb R## is an integrable function in any closed interval ##[x_{i-1}, x_i]## I want to know how can we go on for proving that ##f## is also integrable.

If we can prove that if ##f## is integrable in any closed interval ##[x_{i-1}, x_i]## then we can go on for proving it is integrable on ##[a,b]##. I can see that ##f## and ##g## are just unequal at the end points, but how to use their inequality in points between to prove the integrability from one to the other.

Thank you.

$$

f(x) =g(x)

$$

(##g:[a,b] \mapsto \mathbb R## is an integrable function in any closed interval ##[x_{i-1}, x_i]## I want to know how can we go on for proving that ##f## is also integrable.

If we can prove that if ##f## is integrable in any closed interval ##[x_{i-1}, x_i]## then we can go on for proving it is integrable on ##[a,b]##. I can see that ##f## and ##g## are just unequal at the end points, but how to use their inequality in points between to prove the integrability from one to the other.

Thank you.