Continuity of Functions at a Point: The Role of Addition and Multiplication

[dannysaf]

1)Let f and g be functions such that f (x) + g(x) and f (x) − g(x) are
continuous at x = x₀ . Must f and g be continuous at x = x₀ ?

2)What can be said about the continuity of f (x) + g(x) at x = x₀ , if
f (x) is continuous and g(x) is discontinuous at x = x₀ ?

3)What can be said about the continuity of f (x)g(x) at x = x₀ , if
f (x) is continuous and g(x) is discontinuous at x = x₀ ?

 

[HallsofIvy]

I think the facts that f(x)= ((f+g)(x)+ (f-g)(x))/2 and g(x)= ((f+g)(x)- (f-g)(x))/2 will help a lot!