# The sum of continuous almost everywhere function is also continuous almost everywhere?

My textbook is asking me to solve this exercise. I thought this is trivial, but The author put ‘prove or disprove’. So I’m confusing.

Let f and g be a real-valued function on [a,b]. And assume that f and g are continuous almost everywhere on [a,b]. Prove or disprove : f+g is continuous almost everywhere.

I think this is true. If f is continuous at x and g is continuous at x, f+g is continuous at x. So if A is the set of points of discontinuity of f, and B is of g, then AUB contains the set C of discontinuous points of f+g, i.e. C is contained in AUB

By definition, A and B is of measure zero. And the sum of countably many measure zero sets is also of measure zero. So, AUB is of measure zero. Therefore C is of measure zero. So f+g is continuous almost everywhere.

Am I right?

Thanks

• You're correct. – Math1000 Mar 30 '18 at 6:04
• Oh :) Thanks for checking my answer! – ylh0501 Mar 30 '18 at 6:09