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 \(x = 2.\)

What is continuity? by Eddie Woo

Conclude the lesson by giving your students the classic monk-mountain problem. I like the problem as described here, and the solution as seen here.

Conclude by giving your students these challenges:

- A Jar of Teddies by NRICH
- Who's the Best? by NRICH
- A Lopsided Pyramid

Find the length \(x{:}\)

Here's the solution. There are only two changes I would make to this solution. The first, is to split the base of the first triangle into \(25 - x\) and \(x,\) instead of \(m\) and \(n.\) This gets you the quadratic \(-x^2 + 25x - 144.\) Solving, you get \(x = 9, 16.\) I would then choose \(x = 9,\) because it more closely matches the diagram. The second change I would make, is to use mixed numbers throughout, instead of decimals.

A Weird Calculator by MAA: This problem is pretty difficult. I doubt many students will find success. However, I think the problem is still worth a try. Its solution is very interesting.