Using these 2 facts for integrable functions f, g

$\displaystyle 1. \int_a^b (f+g)=\int_a^b f+\int_a^b g

$

$\displaystyle 2. \int_a^b cf=c\int_a^b f $for a constant $\displaystyle c$

Prove that any integrable function f can be changed at a finite number of points in the interval [a,b] without changing its integrability or its integral.

Start of Proof

I know that I only have to prove it for changing 1 point, and an induction argument will take care of the rest.

But how do I go about doing this?

I started with the definition of integrability

$\displaystyle sup{ }L(f,P)=inf{ }U(f,Q)$, where $\displaystyle P $ and $\displaystyle Q $ are partitions of $\displaystyle [a,b]$

and so dealing with only the sup and a partition $\displaystyle P

a<x_1<b)$ and $\displaystyle Q

a<x_2<b)$

I need to prove

$\displaystyle sup{ }f[a,x_1](x_1-a)+sup{ }f[x_1,b](b-x_1)=sup{ }f[a,x_2](x_2-a)+sup{ }f[x_2,b](b-x_2)

$

How do I use the given facts to solve this problem. Or do I need a new approach

Thanks in advance.