Additivity of riemann stieltjes integral

[gotjrgkr]

Homework Statement

Hi! I'm studying line integral in vector calculus and I've faced a difficulty related with the additivity of line integral.
I really hope to get an answer for my question through this site.

(2.17) Theorem
: If $\int$fd$\phi$ from a to b exists and a<c<b, then $\int$fd$\phi$ from a to c and $\int$fd$\phi$ from c to b both exist and
$\int$fd$\phi$from a to b = $\int$fd$\phi$ from a to c + $\int$fd$\phi$ from c to b.

I refer to this theorem in the text measure and integral by wheeden and zygmund.

The book let's me know the fact such that the converse of the theorem is not true if both f and $\phi$ have discontinuity at c.

What I want to know is to check if following statement is true; [Suppose a<c<b. Assume that not both f and $\phi$ are discontinuous at c. If $\int$fd$\phi$from a to b and $\int$fd$\phi$from c to b exist, then $\int$fd$\phi$from a to b exists and
$\int$fd$\phi$ from a to b = $\int$fd$\phi$from a to c + $\int$fd$\phi$from c to b]

$\epsilon$-$\delta$ notations are used for the definition of riemann stieltjes integral here.

In short, I want to know whether the above statement is true and where the proof of it is written if it exists.
2. Homework Equations [/b]

The Attempt at a Solution

I've tried to prove it by myself and in my proof, I couldn't find the need for the condition 'not both f and $\phi$ are discontinuous at c.'.

Thank you for reading my long questions!
I 'll wait for your reply!

 

[mighty2000]

Dear student,

Thank you for reaching out to us with your question. Line integrals in vector calculus can be quite tricky, so it's great that you are seeking clarification on this topic. To answer your question, yes, the statement you have proposed is true. In fact, it is a consequence of the additivity theorem that you mentioned in your post.

To prove this statement, we can use the definition of Riemann-Stieltjes integral. Let's assume that f and φ are continuous at c. Then, using the additivity theorem, we have:

∫fdφ from a to b = ∫fdφ from a to c + ∫fdφ from c to b

Now, since both ∫fdφ from a to b and ∫fdφ from c to b exist, we can take the limit as c approaches b on both sides of this equation. This gives us:

lim c→b ∫fdφ from a to b = lim c→b ∫fdφ from a to c + lim c→b ∫fdφ from c to b

Since f and φ are continuous at c, we can use the continuity property of the Riemann-Stieltjes integral to rewrite this as:

∫fdφ from a to b = ∫fdφ from a to b + ∫fdφ from b to b

But, since b to b is just the integral of a constant function, it evaluates to 0. Therefore, we have:

∫fdφ from a to b = ∫fdφ from a to c + 0

which simplifies to:

∫fdφ from a to b = ∫fdφ from a to c

This proves that if both f and φ are continuous at c, then the statement you proposed is true. The condition 'not both f and φ are discontinuous at c' is necessary because if both functions are discontinuous at c, then the additivity theorem does not hold. This means that the limit as c approaches b on both sides of the equation would not be equal, and the proof would not hold.

I hope this helps clarify your doubt. If you have any further questions, please don't hesitate to ask. Good luck with your studies!