Proof of Continuity: Homework Statement

[mrchris]

Homework Statement

If the function f+g:ℝ→ℝ is continuous, then the functions f:ℝ→ℝ and g:ℝ→ℝ also are continuous.

Homework Equations

The Attempt at a Solution

Ok, just learning my proofs here, so I'm not sure if my solution is cheating or not rigorous enough. take f(x)= {-1 if x≥0, 1 if x<0} and take g(x)= {1 if x≥0, -1 if x<0}. Then the function (f+g)(x) is a constant function equal to 0 everywhere. since g(x) and f(x) are both discontinuous at x=0, this is a contradiction to the given statement. Basically, I don't know if it is ok to use a piecewise function like this to disprove a statement.

 

[Dick]

Yes. That's perfect. You've found discontinuous functions f and g such that f+g is continuous. So the statement is false.