# Maths Shop Window

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?

Recall that a function is any rule which assigns a unique value in the range of the function to any value in the domain of the function. It is common to encounter functions which can be expressed through simple algebraic equations, but this is not the only way to define functions.

To get you started you might think about these sorts of functions: polynomials, trig functions, exponentials, integer powers of the variable and decide in which function categories these belong.

You might then consider some more complicated functions. For example, some might be defined in two parts such as $f(x) = A(x), x\geq 0$ and $f(x) = B(x), x<0$.

Don't be afraid to be inventive!

### Why do this problem?

This problem introduces students to the concept of different categories of real functions which permeate advanced mathematics. It focuses on understanding the properties of the categories as a whole rather than the properties of individual examples. Hopefully students will leave with the realisation that smooth functions are a very special group of functions along with a wider understanding of functions, continuity and differentiability.

### Possible approach

### Key questions

Do you understand all of the terms?

In what ways can you describe or represent a function?

How would you describe in words the 'essence' of each function category?

### Possible extension

### Possible support