(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Continuity Proof

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