Appropriate context for teaching derivative (undergraduate/graduate)

Question

(Repost from MO, where the question will eventually be closed.)

This question is related to lectures I have to make concerning differential calculus in one variable, but the students are quite advanced, and prepare themselves to be high school teachers.

Requiring a function to be defined on an interval is not sufficient (think of 𝑥↦1/𝑥), nor is it sufficient to only consider open subsets of $\mathbf R$ (think of a function defined on $[0;1]$).

In principle, that no point is isolated suffices to define the derivative, but this may sound as a too far-fetched hypothesis.

To add a bit of context, let me recall that the standard French curriculum for calculus is particularly touchy on hypotheses. For example, there are 3 Taylor formulas (Lagrange, Young, with integral remainder), each with its precise formulation, and, as a consequence, students are supposed to pay attention whether one assumes derivability on the interior only, or everywhere…

So what do you feel is the reasonable context for a function defined over a subset of the reals, to discuss differentiability and higher derivatives of functions in 1 variable?

Answer 1

If you are following a textbook, I would strongly recommend adopting their convention.

If you are writing your own lecture notes, there are several ways I see to proceed.

My personal favorite approach (if your students are sophisticated) would be to have them suggest conditions, challenge them with situations where it seems reasonable for the "derivative" to be defined but which is not covered by their condition, then have them come up with a new condition which covers that case as well. Repeat the process until the entire class (including yourself) are satisfied with the proposed definition.

Alternatively, you could do this yourself and come up with a suitably general definition, and then adopt this as the definition for the course without any student input.

An example definition of this sort:

Let $\Omega \subset \mathbb{R}$, and let $p \in \Omega$ be a limit point of $\Omega$. Let $f: \mathbb{\Omega} \to \mathbb{R}$ be a function. We say that $f$ is differentiable at $p$ iff $$\lim_{x \to p} \frac{f(x) - f(p)}{x-p} \textrm{ exists}$$ In this case we call this limit the derivative of $f$ at $p$, denoted $f'(p)$.

A function $f$ is said to be differentiable on $\Omega$ if it is differentiable at every point of $\Omega$.