I'm teaching a first-year calculus course, that is mid-way between a first intro to university-level calculus, and intro to real analysis (I'm based in Australia, for reference). We assume the intermediate value and extreme value theorems, but do proofs for Rolle's, MVT, l'Hôpital and other elementary theorems. (A slightly less than rigorous definition of limit is given, but I explained the idea of the epsilon-delta definition in words.) I was wondering if anyone has seen examples of a function or functions written down that displays a combination of all the features one might consider illustrating in such a course? A vain hope is that one could have a small collection of functions (no more than two or three) that combine, among them all the features one might want to illustrate over the course.
I'm not sure this would be helpful, but I feel it would be interesting to students to see a more complicated function than familiar elementary functions, but which they gain all the tools to analyse over various intervals in the domain over the course.
EDIT: perhaps I wasn't clear: I would be interested in hearing if people have seen a short list--no more than three or so--of explicit functions displaying as many features that get taught in first-year calculus as possible. I don't mean families or classes of functions. If the answer is no, comments to that effect from people who can say 'I've been teaching for 40 years in country X (and Y and...) and never seen such a thing' would be useful. Or perhaps someone can offer sound grounded and sound practice-driven advice on why such a thing would be useless or even detrimental to students appreciation, even if offered alongside a plethora of small and simpler bite-sized examples.
"Thank you for your consideration"