# Rudin Reference

Hello,

I was wondering where I can find a proof to the following theorem:

If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then

F(x) = int(f,dt,a,x) a<=x<=b.

And the converse.

He gives the theorem are page 324 and a reference in his bibliography. I was wondering where I detailed proof for this theorem.

morphism
Homework Helper
He proves that in his Real & Complex Analysis book. It's Theorem 8.21 on page 169 of the first edition.

Thank you dearly.

Hey morphism,

do you know where I can find the proofs for the theorems 11.23 (a), (d), (e), (f), 11.24(b), 11.26, 11.27, 11.29, and 11.32 extended to Lebesgue integrals of complex functions?

morphism
Homework Helper
Have you tried to prove them yourself? They easily follow from their real-valued analogues.

He just gives the converse to

If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then

F(x) = int(f,dt,a,x) a<=x<=b.

Where can I find the proof for this?

Sorry, the actual theorem is:

If f in L on [a,b] and F(x)=int(f,t,a,x) (a<=x<=b) then F'(x)=f(x) almost everywhere on [a,b].

and he uses strictly eveywhere for continuity, why is this>

morphism
Homework Helper
I'm not sure that I understand what it is you're asking.

He just gives the converse to

If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then

F(x) = int(f,dt,a,x) a<=x<=b.

Where can I find the proof for this?
Like I said, this is in one of his other books (with your typos corrected!), Real and Complex Analysis.

Sorry, the actual theorem is:

If f in L on [a,b] and F(x)=int(f,t,a,x) (a<=x<=b) then F'(x)=f(x) almost everywhere on [a,b].

and he uses strictly eveywhere for continuity, why is this>
Where is this from? And continuity of what is being used?

It was from baby rudin on page 324. I can only find the proof for the converse in Real and Complex Analysis.