When the domain of a function is naturally discrete then the function must be discontinuous.
Good examples are tables of values.
In a lottery, the probability of matching N numbers is a discrete function. Another standard example is postage charges based on whole ounces even though weight is generally considered a continuous domain. Here, I must disagree with jlu's accelera- tion example. Jerky motion is not discontinuous in acceleration. The fact is that we could never solve any problem if we could not idealize it.