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Maths Shop Window
Age 16 to 18
Challenge Level 
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
This problem assumes that students will have encountered informally ideas of continuity and differentiability. You can role play this task, or play it straight as you wish.
Start by asking the students (individually or in groups) to come up with a single example of a function from each category, and then share these collectively. Key issues which are likely to occur here are 1) Some functions are members of multiple categories and 2) students will probably want to sketch examples for some categories.
Once the students are involved with the concept, ask them to produce the 'best' example of a function from each category along with a representation of the category as a whole, as if for a small display poster which would sit in a shop window.
Note: possible representations of the categories are: multiple algebraic examples, sketches/graphs, descriptions in words (such as 'functions which can be drawn without taking the pen off the paper') or formal mathematical descriptions, such as $f(x+Na) = f(x)$.
Once students have worked on their ideas they can share them, where the challenge is for others to guess the category. A 'good' solution to this part will be one which clearly points to one and only one category. Be prepared for multiple suggestions and representations and you might find differences of opinion concerning which solutions are 'best'. You can put some of these on the walls of
your classroom to remind students of the meaning of the different function categories.
The last part of the question is more traditionally algebraic and can be attempted independently if you wish. Simply tabulate a grid with each property as a row and column and ask students to find algebraic examples of functions which fit in the different grid cells. How many grid cells is it possible in principle to fill? Are members of any categories automatically members of other
categories? Do any particularly interesting examples crop up?
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
(Difficult) Ask students to find examples of functions which fit none of these categories. What other sorts of categories of functions would be needed to accommodate their new examples?
Possible support
Focus on curve sketches which best represent each category.
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