I have also observed same question in my book..
If f is a bounded function from [a,b] into the real numbers that is
integrable on [a,b], then for any c such that a < c < b, is f
integrable on [a,c] and is f integrable on [c,b]?
I want to say no, since if I pick , then f is integrable from [-2,2] but not if you move the boundary to 0.
Is that right? Thanks.
So I guess this theorem is true then?
Here is my proof so far:
Let be given. Since f is integrable, then there exists a partition on [a,b] such that
Define and , so we have
Now let , then
Define and
But we know that
I need to show that and
Any hints? Thanks.