1. The problem statement, all variables and given/known data Show that there exist nowhere continuous functions f and g whose sum f+g is continuous on R. Show that the same is true for their product. 2. Relevant equations None 3. The attempt at a solution Let f(x) = 1-D(x), where D(x) is the Dirichlet function Let g(x) = D(x) (f+g)(x) = 1 (f*g)(x) = D(x) - D(x)^2 <-- where I'm befuddled I know that D(x) can be written as the limit of cos(m!*pi*x)^(2n) as n, m --> infinity and that D(x)^2 is then equal to cos(m!*pi*x)^(4n). Since n --> infinity, are D(x) and D(x)^2 equivalent?