Yes. Tying to define f(0) as 0 or 1 won't make it continuous, as approaching from the lines x = 0 and y = 0 will make the limits differ (the definition of lack

Yes. Tying to define f(0) as 0 or 1 won't make it continuous, as approaching from the lines x = 0 and y = 0 will make the limits differ (the definition of lack of a limit).