Students will be given an informal definition of continuity, followed by a formal definition. They will be shown some examples of functions which are not continuous. A good example is

$$f(x) = \dfrac{x^2}{x}$$which is just the identity function with a hole at the origin. You should also show that non-continuous functions can be continuous at particular points. For example,

$$f(x) = \dfrac{1}{x}$$is not continuous at \(x = 0,\) but is continuous at \(x = 2.\) Here's a great video covering all the aforementioned ideas.