Original source of "precise" ε-δ (epsilon-delta) formal definition of a limit?

Question

I frequently see Karl Weierstrass credited for formulating the precise definition of a limit. But what I'd like to know is the origin of the formal definition so common in textbooks, that given a continuous function $f : ℝ → ℝ$, for all $x$ of its domain within the neighbourhood of a constant c (i.e., any point limit of $x$),

\begin{align*} (\lim_{x→\textrm{c}} f(x) = \textrm{L}) &≜ ∀\, ε \in ℝ_{>0} : ∃\, δ \in ℝ_{>0} : ∀\, x ∈ \textrm{DOMAIN}\ f : \\ &\qquad 0 < \left|x - \textrm{c}\right| < δ ⇒ \left|f(x) - \textrm{L}\right| < ε \end{align*}

The Stanford Encyclopedia of Philosophy's "Continuity and Infinitesimals" credits Weierstrass but cites Heine, and then only for the formal definition of a $\textit{continuous}$ function.

Anyone know more history on this or Weierstrass's original publication? Did he use Gottlob Frege's notation or some other kind?

Answer 1

• In the paper On the history of epsilontics by G.I. Sinkevich we can read:

  Unfortunately, Weierstrass himself had never published or edited his lectures. In most cases, they came down to us in notes of his students.

  ...

  The earliest known Weierstrass’ text where the $\varepsilon$-$\delta$ technique is mentioned are differential calculus lecture notes made at a lecture read in the summer term of 1861 in Königlichen Gewerbeinstitut of Berlin. “The lecture notes were made by Weierstrass’ student, H.A. Schwarz, and are now kept in Mittag-Leffler Institute in Sweden. Schwarz was 18 then, and he wrote these notes solely for himself, not to be published” [Yushkevich, 1977, p. 192].

  Schwarz’ notes were found and published by P. Dugac [Dugac, 1972]. It is in these notes that the definition of continuous function in the language of epsilontics appears for the first time:

  “If $f(x)$ is function $x$ and $x$ is a defined value, then, on conversion of $x$ into $x+h$, the function will change and will be $f (x+h)$; difference $f (x+h) – f (x)$ is used to be called the change received by the function by virtue of the fact that the argument convers from $x$ to $x + h$. If it is possible to determine such boundary $\delta$ for $h$ that for all values of $h$, the absolute value whereof is still smaller than $\delta$, $f(x+h) – f(x)$ becomes smaller than any arbitrarily small value of $\varepsilon$, then infinitesimal changes of function are said to correspond to infinitesimal changes of argument. Because a value is said to be able to become infinitely small, if its absolute value can become smaller than any arbitrary small value. If any function is such that infinitesimal changes of function correspond to infinitesimal changes of argument, then it is said to be a continuous function of argument or that it continuously changes along with its argument” [Yushkevich, 1977, p.189].

• Yushkevich A. 1977. Chrestomatija po istorii matematiki. Matematicheskij analis (Reading book on the history of mathematics. Analysis), edited by A.Yushkevich.- Moscow: Prosveschenije. – 224 p.

• Dugac, P. 1972. Elements d`analyse de Karl Weierstrass. – Paris.