Give an example of a set X and a function f: X -> Real numbers such that:
1) f is continuous at 2 points but is discontinuous everywhere else on X.
I have put this:
f(x)=
x^2-x where x belongs to rationals
0 where x doesn't belong to rationals on X= real numbers
Is this right?
2) f is continuous on X and sup {f(x)|x belongs to X} belongs to {f(x)|x belongs to X} but inf{f(x)|x belongs to X} doesn't belong to {f(x)|x belongs to X}
Could this be a possible answer? :
f(x) = x^2 on X = (-10,10]
3) f is bounded on X but not continuous on X
I'm not sure what to put here.