Proof using Riemann Integral definition

[KPutsch]

Homework Statement

Suppose that f:[a, b] → ℝ is a function that is zero for all x ∈ [a, b] except for the values x_1,x_2,…,x_k. Find ∫[a b](f(x)dx) and prove your result.

Homework Equations

Definition of a Riemann integrable function: http://en.wikipedia.org/wiki/Riemann_integral#Riemann_integral

The Attempt at a Solution

I'm simply not sure how to define the tags of the partition. If I let t1 be in [a, x1), t2 in (x1, x2), ..., tn in (xk, b], then the Riemann sum will be zero, but I'm not making use of the fact that f(xi) != 0. This is where I'm stuck, how do I make use of the fact that there is a finite set of discontinuities in setting up this proof?

 

[╔(σ_σ)╝]

Select your partition to be really small so that the non zero points can be made smaller than epsilon. Alternatively, you could just say your set of discontinous points have measure 0.

 

[KPutsch]

I simply don't know how to do that. I'll let the norm of the partition be less than delta=epsilon, but when working with the Riemann sum, and when t_i = x_i, I'll be left with f(x_i)(x_i - x_i-1) < f(x_i)*epsilon.

Since I don't know what f(x_i) is, I can't put a bound on it. I don't know what else I could let delta be, because what if there are two tags that fall on an x_i and x_i+1, the the Riemann sum will be f(x_i)(x_i - x_i-1) + f(x_i+1)(x_i+1 - x_i) < [f(x_i)+f(x_i+1)]*epsilon.

I really don't like the way this this real analysis class is taught. I'm given definitions with no examples, and then asked to solve problems like this, with no idea of what I'm doing.