Re: Continuity on rationals?
An even more interesting scenario on the properties f(x) can have is the following :
Let g(x) = sin(x) which is continuous over R
Let f(x) = g(x) for x rational (f agrees with g at rationals)
= -g(x) for x irrational (reflection of graph of g along the X axis at irrational x values)
It is easy to prove that f(x) is continouus only at the points K*pi, K belonging to Integers
and discontinuous everywhere ! Similarly, we can construct counter-intuitive functions continuous, say at Integers and
discontinuous elsewhere ! The old classic ,Hobson's "Theory of Functions-I,II" would be an illuminating read !!!!