Proof of Continuity of f+g & f*g on R

[ƒ(x)]

Homework Statement

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.

Homework Equations

None

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?

 

[LCKurtz]

Write out the definitions of f(x) and g(x) in terms of x and look at their product.

 

[ƒ(x)]

Write out the definitions of f(x) and g(x) in terms of x and look at their product.

You mean it will alternate between 0^2 and 1^2 then?

 

[LCKurtz]

You mean it will alternate between 0^2 and 1^2 then?

No. I mean if x is rational what are f(x) and g(x) and their product. And what if x is irrational?