f is defined on domain [0, 1]. If x = 1/n where n is a positive integer, then f(x) = 1. Otherwise, f(x) = 0. Can I check, how many values of x in the interval [0,1] exist such that f is continuous at x?
Yes, you can. Hint: There are three types of points: $x=0, x=\frac1n$ and all other points. How many are there of each type? At which of those is $f$ continuous?