That's because you are thinking of continuous functions. You have actually gave an example of a non-continuous one, it's just that you think of it as two functi

That's because you are thinking of continuous functions. You have actually gave an example of a non-continuous one, it's just that you think of it as two functions connected with input conditional. Other replier already gave you some common example, but let me add another one: signum(x), which returns if number is negative, positive or zero.