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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

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

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