# $\epsilon-\delta$ definition of continuity

According to Wikipedia Bolzano and later Weierstrass were the first who gave an $\epsilon-\delta$ definition of continuity and convergence. But did they already use the letters $\epsilon$ and $\delta$ in their formulation and, if yes, did they have any particular reason for their choice?

Regarding the use of $\epsilon$ and $\delta$ in the context of the definition of continuity of functions, we can say that they "were in the air" since Cauchy.

See Augustin-Louis Cauchy's definition of limit in terms of infinitesimals in his Cours d'Analyse (1821):

As we can see, the definition is purely "verbal".

See page 26 for continuity :

Let $f(x)$ be a function of the variable $x$, and suppose that for each value of $x$ between two given limits [i.e. in an interval $[x_0,x_1]$], the function always takes a unique finite value. If, beginning with a value of $x$ contained between these limits, we add to the variable $x$ an infinitely small increment $\alpha$, the function itself is incremented by the difference

$$f(x+\alpha)− f(x),$$

which depends both on the new variable a and on the value of $x$. Given this, the function $f(x)$ is a continuous function of $x$ between the assigned limits if, for each value of $x$ between these limits, the numerical value of the difference

$$f(x+\alpha)− f(x)$$

decreases indefinitely with the numerical value of $\alpha$. In other words, the function $f(x)$ is continuous with respect to $x$ between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself.

We can see in page 22, in the context of the discussion about infinitely small and infinitely large quantities, usually denoted with $\alpha$ :

if $k$ denotes a finite quantity different from zero and $\epsilon$ denotes a variable number that decreases indefinitely with the numerical value of $\alpha$, the general form of infinitely small quantities of the first order is

$$k \alpha \ \text {or at least} \ k \alpha (1 ± \epsilon).$$

Regarding $\epsilon$ and $\delta$ occurrences, see A.-L.Cauchy, Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal, Paris, 1823; in Oeuvres, series 2, vol.4, page 44 :

If $f$ is a continuous function between the limits $x=x_0$ and $x=X$ [...].

Let $\delta, \epsilon$ be two very small numbers, the first is chosen so that for all numerical values of $i$ less than $\delta$ and for any value of $x$ between the limits $x_0, X$, the ratio

$$\frac {f(x+i)− f(x)}{i}$$

will always be greater than $f'(x) - \epsilon$ and lesser than $f'(x) + \epsilon$.

For Bernard Bolzano, see the English translation (by S.B.Russ) of Bolzano's paper, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Prague 1817), page 162 :

According to a correct definition, the expression that a function $fx$ varies according to the law of continuity for all values of $x$ inside or outside certain limits means just that: if $x$ is some such value, the difference $f(x + \omega) - fx$ can be made smaller than any given quantity provided $\omega$ can be taken as small as we please.

Tralles described discontinuity as a “jump” in the values of the function (Fol. 10v), but he also gave an algebraic description, in an early use of the letters $\delta$ and $\epsilon$ , by which he stated that “for however small a number $\delta$ of the radical $\Delta$” (he considers the function to be dependent on an increase $\Delta$) “and for the value $\alpha$ of the function corresponding to $\delta$ the inequality $\alpha^{\epsilon} > (1 - \omega)^{\epsilon}$ must hold” (Fol. 19v).
• Mauro, it would be nice if you could make it clear that Cauchy never gave an epsilon-delta definition of continuity, but on the contrary gave an infinitesimal definition. Namely, an infinitesimal $x$-increment always leads to an infinitesimal $y$-increment. – Mikhail Katz Apr 20 '17 at 13:46
• Mauro, describing Cauchy's definition of continuity as "verbal" is ambiguous. Taken literally this is not correct, since Cauchy does use formulas in his definition: for an infinitesimal $\alpha$, he requires $f(x+\alpha)-f(x)$ to be infinitesimal. So this definition is not "purely verbal" since it uses formulas. If what you mean is Grabiner's dichotomy of "verbal" definitions as opposed to epsilon-delta ones requiring alternating quantifiers, then it is of course true that the infinitesimal definition requires no alternating quantifiers. But that is its strength rather than weakness. – Mikhail Katz Apr 23 '17 at 9:30
See the answer at https://mathoverflow.net/questions/82302/why-do-we-use-epsilon-and-delta. In particular, $\varepsilon$ is from the first letter of the French word "erreur" (error). Probably $\delta$ is from the first letter of the French word "différence."