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]
$\displaystyle U(P,f)-L(P,f)=\sum_{i=0}^{N}[f(x_M)-f(x_m)][x_{i+1}-x_{i}]$
Where $\displaystyle x_M$ is the max on each sub interval
$\displaystyle x_m $ is the min on each sub interval
They exsist by the extreem value theorem.
Then on each subinterval by the MVT there exists a $\displaystyle c \in (x_{i},x_{i+1})$ such that
$\displaystyle f(x_m)(x_{i+1}-x_i)=F(c_i) \mbox{ and } f(x_M)(x_{i+1}-x_i)=F(c_{i+1}$
so we get
$\displaystyle U(P,f)-L(P,f)=\sum_{i=0}^{N}[f(x_M)-f(x_m)][x_{i+1}-x_{i}]=\sum_{i=0}^{N}F(c_{i+1})-F(c_i)$
$\displaystyle =F(c_{N+1})-F(C_0)=F(b)-F(a)$
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 $\displaystyle U(f,p)-L(f,p)< \epsilon$
let $\displaystyle \epsilon > 0$ 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 $\displaystyle \delta > 0$ such that for $\displaystyle |x-y|< \delta \implies |f(x)-f(y)| < \frac{\epsilon}{b-a}$
Choose a partition P of [a,b] such that the mesh (each subdivsion) is less that $\displaystyle \delta$ $\displaystyle P=\{ a=x_0,x_1,...x_{N+1}=b \}$
Now by the extreem value theeorem and the continuity of f in each subinterval $\displaystyle [x_i,x_{i+1}]$ there exists points
$\displaystyle x_m,x_M$ such that $\displaystyle f(x_m)$ is a min and $\displaystyle f(x_M)$ is a max on each sub interval
Then for this partition note that since the mesh is less that delta $\displaystyle f(x_M)-f(x_m) < \frac{\epsilon}{b-a}$
$\displaystyle U(f,p) -L(f,p)=\sum_{i=0}^{N}(f(x_M)-f(x_m))(x_{i+1}-x_1)< \frac{\epsilon}{b-a}\sum_{i=0}^{N}(x_{i+1}-x_i)=$
$\displaystyle \frac{\epsilon}{b-a}(b-a)=\epsilon$
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?