For homework we had the following problem which I actually found difficult and fun. (Ross, Section 32, Problem 7).
"Let f be an integrable function on [a,b] and g be a function which agrees with f except for finitely many points, show that g is also integrable and furthermore:
INT(a,b;f) = INT(a,b;g)"
My book (in the back) gives a sketch of the solution. Which is a nightmare.
I came up with two solutions, one of which the professor did in class. But I did not like it. I will explain why. He used the following theorem:
Theorem: If f and g are integrable on [a,b] then f+g is too and furthermore:
INT(a,b;f)+INT(a,b;g) = INT(a,b;f+g)
The reason why I did not like it is because it uses a theorem which was not proved, i.e. from section 33.
See if you propose another proof without that. I found it fun. Note: We use the Darboux definition of the integral.