In order for the limit to exist we need both one-sided limits to exist.
As x approaches zero from the left, y gets close to 0 also, therefore
\lim_x\to 0^-f(x) = 0.
As x approaches zero from the right, y gets close to 1, therefore
\lim_x\to 0^ + f(x) = 1.
Since the left-side limit is 0 and the right-side limit is 1, and 0 and 1 aren't the same, therefore
does not exist.