Proof that f(x) = 1/sqrt(x) is Riemann Integrable

[Vespero]

Homework Statement

The problem given is:

Show that the function f(x) = 1/sqrt(x) is integrable on the compact interval [0,1].

Homework Equations

We are only allowed to use theorems, definitions, and properties that have been covered in class or are in the book. The ones I believe to be relevant are the following:

Definition of Riemann Integral:
Let f be defined on the compact interval [a,b].
Then, if lim(delta x --> 0) sum(f(x)delta x exists, it is called the Riemann integrable.

We know that if lim (Upper Sum) = lim (Lower Sum), then the function is Riemann integrable.

Theorem: If f is continuous on a compact interval [a,b], then f is Riemann Integrable.

Theorem: If f is continuous over a compact interval [a,b], except for countably infinitely many points, then f is still Riemann integrable.

The Attempt at a Solution

So far, I have looked at the Darboux Sum Theorem, if that's what it's called, but am not quite familiar with it yet and can't find a way to express the sums properly and show that they converge to the same limit. I have also considered that the function is continuous on [0,1] except at the single value 0. However, does f have to be on the interval for this to be true? Is it true if one of the discontinuities is on an endpoint? If valid, would I need to show that f is continuous on (0,1] and thus state that it is continuous on [0,1] except for at countable points?

Many thanks for all your help!

 

[Office_Shredder]

I would be surprised if you were required to prove that this is a continuous function on (0,1). Remember the definition only requires you to examine the limit as the lower bound goes to zero, so the fact that the function isn't defined at 0 isn't a problem.

It seems like using the fundamental theorem of calculus to find the anti-derivative would be a good place to start

 

[Vespero]

What do you mean by the lower bound here? Are you referring to the value of the function at the right-most end of the subinterval--in this case, the minimum function value in the subinterval?

As far as I can tell, we haven't covered the fundamental theorem of calculus in this class yet, as it's not in my notes and I don't remember having covered it, and I pay attention and haven't skipped a class. Thus, we aren't allowed to use it.

 

[Vespero]

So, I could possibly prove Riemann integrability simply from the definition? Or is an anti-derivative necessary?