Should the notion of continuity, usually ascribed to Cauchy, be ascribed to Leibniz?

Question

In his text, Deleuze and the History of Mathematics, Simon Duffy writes:

Leibniz also thought the following to be a requirement to continuity: "When the difference between two instances in a given series, or in whatever is presupposed, can be diminished until it becomes smaller than any given quantity whatever, the corresponding difference in what is sought, or what results, must of neccesity also be diminshed or become less than any given quantity whatever".

Leibniz, Philosophical Papers and Letters, 2nd ed, Ed & Trans. L.Loemker

This sounds very close to the traditional epsilon & delta method of undergraduate analysis which I'd been taught had originated with Cauchy.

How far should Leibniz be credited with the idea of continuity as traditionally understood in mathematics?

Answer 1

"the traditional epsilon & delta method of undergraduate analysis which I'd been taught had originated with Cauchy": you have been taught a misconception. Cauchy defined continuity as follows: if $\alpha$ is infinitesimal then $f(x+\alpha)-f(x)$ is also infinitesimal. See Cours d'Analyse. Notice that there are no epsilons or deltas here, only infinitesimals. Cauchy consistently gave this definition of continuity starting in his 1821 textbook until as late as his 1853 paper clarifying the hypothesis of his "sum theorem"; see this article. As you noticed, Leibniz authored very similar formulations but did not really think of this as a property of functions (which weren't introduced until Euler). More on Leibniz here.

For a recent paper trying to set the record straight on Cauchy you can read this.

Answer 2

If you are just interested in the "epsilon-delta" idea (rather than continuity per se)... Archimedes "method of exhaustion" may qualify.

LINK

Answer 3

I don't think that the notion of continuity should be ascribed to Cauchy alone; here's a picture from the Cours d'Analyse

Page 34

Page 35

This is really vague, because it relies on the concept of infinitesimal. Of course this is the idea of continuity in the modern sense, but only when we change it into what Weierstraß precisely defined in terms of $\varepsilon$-$\delta$. Also Heine should be mentioned, with continuity defined in terms of convergence of sequences.

Leibniz cannot be credited: he didn't even think to cases where functions are not continuous.