Prove that if f(x) is continuous on a closed interval [a,b] then it is integrable there.
I need some help with this one guys. Thanx.
Is Anyone Up to the challenge?
Prove that if f(x) is continuous on a closed interval [a,b] then it is integrable there.
I need some help with this one guys. Thanx.
Is Anyone Up to the challenge?
Here is the basic Idea
Let P be any partion of [a,b]
Where is the max on each sub interval
is the min on each sub interval
They exsist by the extreem value theorem.
Then on each subinterval by the MVT there exists a such that
so we get
This is just a sketch you will need to justify and fill in the details
TES
I will leave the above proof here but it is NOT what you asked for. I should have read the question more carefully. That is my mistake.
To show that this is integrable we need to show that there exists a partition of [a,b] such that
let be given
Since f is continous of a closed bounded (i.e compact ) interval f is uniformly continous.
Now by the uniform cont. of f there exists a such that for
Choose a partition P of [a,b] such that the mesh (each subdivsion) is less that
Now by the extreem value theeorem and the continuity of f in each subinterval there exists points
such that is a min and is a max on each sub interval
Then for this partition note that since the mesh is less that delta
Now, can I apply this (by epsilon-delta proof) to the function, let's say... 2x, and show that this function is integrable?
I'm kind of at a loss, because in most cases of the epsilon-delta proofs of ordinary limits of functions ( as in taking the derivative for instance), the limit L is either known, or can be found through algebraic methods. But, in the case of integration, The value of the integral must remain a mystery until the function has, in fact, been integrated.
Now, if I choose an epsilon, how can I show that the Riemann Sum is approaching some value A without having previous knowlege of what that value may be?
Why does this stuff make my head spin?