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Limits, convergence, and continuity

Beginning perhaps with D'Alembert, it was an oft-repeated statement by 18th century mathematicians that the calculus should be "based on limits." It is not surprising then that the arithmetization program culminated in the establishment of the concept of the limit and of those other fundamental concepts that were connected with it, including convergence and continuity.

Bolzano and Cauchy gave similar definitions of limits, convergence, and continuity. They were contemporaries, "both chronologically and mathematically."

Limits

D'Alembert's own definition of limit was as follows:

... the quantity to which the ratio $z/u$ approaches more and more closely if we suppose $z$ and $u$ to be real and decreasing. Nothing is clearer than that.

Bolzano and Cauchy both developed (independently) a concept of limit that was an advance over D'Alembert's and all previous attempts:

• it was free from the ideas of motion and velocity and did not depend on geometry
• it did not retain the (unnecessary) restriction, that a variable could never surpass its limit

For example, Cauchy's definition was constructed using only these three elements

• the variable
• the limit
• the quantity by which the variable differed from the limit

and stated simply that the variable and its limit differed by less than any desired quantity, as follows:

When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last [latter fixed value] is called the limit of all the others [successive values].

The effect of this definition was to transform the infinitesimal from a very small number into a dependent variable. Cauchy put this as follows:

One says that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge toward the limit zero.

Cauchy's definition is wholly verbal, although it has been noted elsewhere that he translated such statements into the precise language of inequalities when he needed them for proofs. Even so, it was Weierstrass who finally provided a formal $\delta,\varepsilon$ definition of limit. His student Heine published this definition of the limit of a function using notes from Weierstrass's lectures:

$\displaystyle \lim_{x \to \alpha}f(x) = L$ if and only if, for every $ε > 0$, there exists a $δ > 0$ so that, if $0 < |x - a| < δ$, then $|f(x) - L| < ε$. ===='"`UNIQ--h-2--QINU`"'Convergence==== Working with the notion of a sequence that "converges within itself," Bolzano and Cauchy sought to relate the concepts ''limit'' and ''real number'', somewhat as follows: :If, for a given integer $p$ and for $n$ sufficiently large, $S_{n+p}$ differs from $S_{n}$ by less than any assigned magnitude $\varepsilon$, then $S_{n}$ also converges to the (external) real number $S$, the limit of the sequence. Meray understood the error involved in the circular way that Bolzano and Cauchy had defined the concepts ''limit'' and ''real number'':'"`UNIQ--ref-00000006-QINU`"' * the limit (of a sequence) was defined to be a real number $S$ * a real number was defined as a limit (of a sequence of rational numbers) To avoid this circularity, Meray avoided references to convergence to an (external) real number $S$. Instead, he described convergence using only the rational numbers $n$, $p$, and $\varepsilon$, which is the Bolzano-Cauchy condition. Weierstrass also understood the error involved in earlier ways of defining the concepts ''limit'' and ''irrational number'':'"`UNIQ--ref-00000007-QINU`"' * the definition of the former presupposed the notion of the latter * therefore, the the definition the latter must be independent of the former ===='"`UNIQ--h-3--QINU`"'Continuity==== Bolzano saw that the intermediate value theorem needed to be proved "as a consequence of the definition of continuity." In his 1817 proof, he introduced ''essentially'' the modern condition for continuity of a function $f$ at a point $x$:'"`UNIQ--ref-00000008-QINU`"' :$f(x + h) − f(x)$ can be made smaller than any given quantity, provided $h$ can be made arbitrarily close to zero The caveat ''essentially'' is needed because of his complicated statement of the theorem, as noted above. In effect, the condition for continuity as stated by Bolzano actually applies not at a point $x$, but within an interval. In his 1831-34 manuscript, Bolzano provided a definition of continuity at a point (including one-sided continuity). However, as noted above, this manuscript remained unpublished until eighty years after Bolzano's death and, consequently, it had no influence on the efforts of Weierstrass and others, who completed the arithmetization program.'"`UNIQ--ref-00000009-QINU`"' In 1821, Cauchy added to Bolzano's definition of continuity at a point "the final touch of precision":'"`UNIQ--ref-0000000A-QINU`"' :for each $\varepsilon > 0$ there is a $\delta > 0$ such that $|f(x + h) − f(x)| < \varepsilon$ for all $|h| < \delta$ Here it's important to note that, as he stated it, Cauchy's condition for continuity, alike with Bolzano's, actually applies not at a point $x$, but within an interval.'"`UNIQ--ref-0000000B-QINU`"' Once again, it was Weierstrass who, working very long after both Bozano and Cauchy, formulated "the precise $(\varepsilon,\delta)$ definition of continuity at a point."

Primary sources

• Heine, E., "Die Elemente der Funktionenlehre," Journal fur die Reine und Angewandte Mathematik, 74 (1872), 172-188.

References

• Bogomolny, A. "What Is Calculus?" from Interactive Mathematics Miscellany and Puzzles http://www.cut-the-knot.org/WhatIs/WhatIsCalculus.shtml#Alembert, Accessed 27 May 2014
• Dunham, W. (2008). The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton University Press.