Using these 2 facts for integrable functions f, g

for a constant

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

, where

and

are partitions of

and so dealing with only the sup and a partition

a<x_1<b)" alt="P

a<x_1<b)" /> and

a<x_2<b)" alt="Q

a<x_2<b)" />

I need to prove

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

Thanks in advance.