Define f : R → R by f(x) = 5x if x is rational and f(x) = x² + 6 if x is irrational. Prove that f is discontinuous at 1 and continuous at 2. Are there any other point besides 2 at which f is continuous?
at x=1 any open neighborhood about it will contain both rational and irrational numbers, but the irrational part is converging to 7 while the rational part is converging to 5 hear.
however at 2 both parts converge to 10.
to find the other point consider where
3 might be another good place to look.
To prove that is discontinuous at 1, let Then for any the interval contains irrational numbers. If is one of them, then
To prove that is continuous at 2, let be given. Then if consider in the interval If is rational, we have If is irrational, then
There is one other point at which is continuous, and that is found by solving the equation for