OME 1968 Problem 2

Justify if continuity can be affirmed, denied or cannot be decided in the point $x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases .

a) It is known only that for all natural $n$ : $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$ .

b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$ .

c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$ .