This is probably the last of three articles on how piecewise-linear functions could be used as a helpful on-ramp to the big ideas in calculus. In the first article, we saw how it’s possible to ...

But I argued that it would be more helpful to students’ understanding of the derivative to start with a simpler case first, namely to use only piecewise-linear functions at the beginning.

We show that for every 0 < s ≤ 1 < p the piecewise linear map fs,p on the interval [0, 1] given by an expanding branch px for 0 ≤ x ≤ 1/p, and a contracting branch sx − s/p for 1/p < x ≤ 1, is ...