You work in a maths shop in the lobby of the Hilbert Hotel.

The manager wants to make a window display to highlight the different types of real valued functions of the real numbers that she has on offer, with the 'best' examples of functions from each category along with a representation which most sums up the 'essence' of each category. She decides that she wants to showcase 9 particularly important types of function categories:

1 Periodic

2 Tending to a vertical asymptote

3 Discontinuous somewhere

4 Decreasing

5 Bounded

6 Infinitely differentiable at all points

7 Singular somewhere

8 Taking finitely many values

9 Unique tangent exists at all points

Think of a few examples of functions from each category and the different ways that you might represent the different categories. What would be the clearest examples and representations that you could think of to showcase these function categories?

It might be that you are in competition with another assistant to produce the best display; if so you will need to convince the manager that your selection of 9 functions and representations is the best; it may be that you will need to work collaboratively simply to dream up any examples in some of the categories! It might be that you wish to suggest a better set of function categories.

Imagine now that you are faced with fussy customers who are likely to request simple examples of functions satisyfing pairs of these properties. Which requests can you satisfy? Which requests will it be impossible to satisfy?