Topological Calculus: away with those nasty epsilons and deltas

Continuous functions[figure from Olver (2022a).

A new approach to calculus has recently been developed by Peter Olver of the University of Minnesota. He calls it “Continuous Calculus” but indicates that the name “Topological Calculus” is also appropriate. He has provided an extensive set of notes, which are available online (Olver, 2022a)].

Motivation

Students embarking on a university programme in mathematics often feel “bombarded” by epsilons and deltas. For nearly 200 years, mathematical analysis has been based upon a formalism introduced by Bolzano, Cauchy and Weierstrass. To get around the difficulties associated with infinitesimal quantities, they defined limits, continuity, derivatives and integrals using the ${\varepsilon-\delta}$ technique.

A real-valued function ${f(x)}$ has a limit ${A}$ at ${x_0}$ if, given any positive ${\varepsilon}$ , however small, a positive number ${\delta>0}$ can be found so that, whenever ${0<|x-x_0|<\delta}$ we have ${|f(x)-A|<\varepsilon}$ . If that is all Greek to you, then you are not alone. Let me make it worse: ${A}$ is the limit of ${f(x)}$ at ${x=x_0}$ provided that

$\displaystyle \forall\varepsilon>0\ \exists\delta>0 : 0<|x-x_0|<\delta \implies |f(x)-A|<\varepsilon \,.$

It is not hard to imagine the perplexity of a young student seeing this for the first time.

Augustin-Louis Cauchy in 1821 and, later, Karl Weierstrass, formalized the definition of the limit of a function which became known as the ${(\varepsilon, \delta)}$ definition of limit. However, in 1817, Bernard Bolzano had introduced the basic idea of the epsilon-delta technique, although his work was not recognised during his lifetime.

A Better Way

The starting point for Olver was the idea that motion is continuous: if something moves from ${A}$ to ${B}$ , it must pass along a continuous path between these two points. This is an accessible and intuitively obvious idea (unless you are Zeno!). Olver presents the arguments in an article in The Mathematical Intelligencer (Olver, 2022b).

Topological continuity is defined thus: A function is continuous if and only if the inverse image of any open set is open. This characterisation is more general, yet simpler, than the limit-based definition usually used in calculus. Olver realised that he could develop basic calculus using this idea as the starting point. He saw that he could construct definitions of derivatives, integrals and convergence in a rigorous way while eliminating all references to epsilons and deltas. In his development, continuity is the basic concept. Limits are inessential but are introduced for convenience (defining them in terms of continuity). The definition of the derivative of a function follows in a straightforward manner.

Olver starts with real valued functions on the real line. A topology ${\mathcal{O}}$ on ${\mathbb{R}}$ is constructed from unions of open intervals. The topological definition of continuity can easily be motivated by considering the inverse images of open inervals. Students will need to learn about the topological properties of the real line. However, they require such knowledge for further progress in mathematics, and usually learn about it in parallel with elementary analysis.

Continuity

The definition of topological continuity originates with Felix Hausdorff (1914):

Definition. A function ${f : \mathbb{R}\rightarrow\mathbb{R}}$ is continuous if, whenever ${S \subset \mathbb{R}}$ is open, then ${f^{-1}(S) \subset\mathbb{R}}$ is open.

From this, Olver proves the usual properties of continuous functions: the functions ${f(x) \pm g(x)}$ , ${f(x)g(x)}$ and ${f(x)/g(x)}$ are continuous ( ${g(x)\ne 0}$ in the last case). He then proves the Intermediate Value Theorem, the Bolzano-Weierstrass Theorem, the Extreme Value Theorem and the Heine-Borel Theorem.

Limits, which are inessential but convenient, are defined as follows. Suppose ${f : I_a \rightarrow \mathbb{R}}$ is continuous on the punctured interval ${I_a : \mathbb{R}\setminus\{a\}}$ . We say that ${f(x)}$ has limiting value ${z \in \mathbb{R}}$ at ${x = a}$ if

$\displaystyle \hat f: I \rightarrow R \qquad\mbox{given by}\qquad \hat f(x) = \begin{cases} f(x), &x \ne a \\ z, &x = a \end{cases}$

is continuous. In that case, we write ${\lim_{x\rightarrow a} f(x) = z}$ . Olver then shows that the continuity characterization of limits includes the limiting behavior of a sequence of real numbers ${\{f_1, f_2, f_3, \dots \}}$ .

Differentiation

Now the definition of derivatives can be given using continuity:

Definition. Let ${f: \mathbb{R} \rightarrow \mathbb{R}}$ be continuous. Then ${f(x)}$ has a derivative at ${a \in \mathbb{R}}$ if there exists ${z \in \mathbb{R}}$ such that the difference quotient function

$\displaystyle q(x) = \begin{cases} \displaystyle{\frac{f(x)-f(a)}{x-a}}, &x \ne a \\ z, &x = a \end{cases}$

is continuous. In this case, we write ${f^\prime(a) = z}$ or, equivalently,

$\displaystyle f^\prime(a) = \lim_{x\rightarrow a} \frac{f(x)-f(a)}{x-a} \,.$

Olver points out that this idea dates back to the Greek mathematician Constantin Carathéodory in the early twentieth century. He then derives all the usual formulas of elementary calculus: the derivatives of sums, products and quotients of differentiable functions, and the explicit form for the derivatives of some algebraic and trigonometric functions. Rolle’s Theorem and the Mean Value Theorem are then proved. Higher order derivatives and Taylor’s theorem are easily handled within the formalism of this approach.

Olver now moves to the topology of the plane, and discusses continuous functions, uniform convergence and partial derivatives in this context. He concludes with a few chapters on integration, arriving at the Fundamental Theorem of Calculus.

Conclusion

The continuity approach to calculus is attractive from a pedagogical perspective: it simplifies the proofs of many important theorems, in particular those involving differentiation. As Olver observes, “the distractions and complications of epsilons and deltas completely disappear without any sacrifice in rigor or generality!”

The student needs to spend some time on the topology of the real line and the plane, but this is fundamental, and well worth learning. Olver concludes that a more accurate title of his approach would be “Topological Calculus”, since it is essentially topological and independent of a choice of metric. It will be interesting to learn how the approach works in the real-world context of a class room.

Sources

${\bullet}$ Hausdorff, Felix, 1914: Grundzüge der Mengenlehre, Verlag Veit & Co, Leipzig. 476pp.

${\bullet}$ Olver, Peter J., 2022a: Continuous Calculus. Lecture Notes, Univ. Minnesota, 131pp. PDF.

${\bullet}$ Olver, Peter J., 2022b: Motion and continuity, Math. Intelligencer, 44, 241–249. Preprint PDF.

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