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Limits, convergence, and continuity

Beginning perhaps with D'Alembert, it was an oft-repeated statement of 18th century mathematics that the calculus could be "based on limits." His 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.

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, namely, continuity and convergence.

Limits

Bolzano and Cauchy developed (independently) a concept of limit that had several advances over 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

Cauchy's definition, in particular, stated only 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 is called the limit of all the others.

Cauchy used this definition to define the infinitesimal as a dependent variable, thus freeing it from previous understandings of it as a very small fixed number:

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.

Weierstrass, in his lectures, provided an $\delta,\varepsilon$ definition of limit, which was presented by Heine as follows:

$\displaystyle \lim_{x \to \alpha}f(x) = L$ 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-00000004-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-00000005-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-00000006-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-00000007-QINU`"' Bolzano and Cauchy gave similar defnitions of limits, derivatives, continuity, and convergence. They were contemporaries, "both chronologically and mathematically."'"`UNIQ--ref-00000008-QINU`"' In 1821, Cauchy added to Bolzano's definition of continuity at a point "the final touch of precision":'"`UNIQ--ref-00000009-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-0000000A-QINU`"' Weierstrass, 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.