# What is the correct statement of Cauchy’s erroneous theorem on continuity?

I read recently that Cours includes a famous, or perhaps infamous, error in that Cauchy states and proves a false result concerning sequences of continuous functions. (Here, obviously, continuous function means continuous according to Cauchy’s intuitive definition of continuity and his notion of the continuum together with Cauchy’s concept of function.)

There appear to be conflicting accounts of what Cauchy actually stated as a theorem.

Some interpret the result as saying that the pointwise limit of a sequence of continuous functions is a continuous function. Let’s call this the "limit interpretation": $f(x) = \lim_{n \to \infty} f_{n}(x)$.

Others say, no, no, no - Cauchy is talking about the series whose terms are a sequence of functions, and that interpreted this way the result is much more “plausible” according to Cauchy’s own definition of continuity - more plausible in the sense that counter-examples are not completely trivial. Let’s call this the "series interpretation": $f(x) = \sum_{n=1}^{\infty} f_{n}(x)$.

A recent translation (see TL/DR below) suggests that the series interpretation is correct, however it is not clear if this translation is a statement of Cauchy’s original 1821 theorem or a statement of his 1853 clarification.

Question: According to Cauchy’s notion of function and the continuum, what is the correct interpretation of Cauchy’s original 1821 theorem and his subsequent 1853 clarification.

TL/DR

Wikipedia states only the limit interpretation, referring to “some” historians and “purported” counter-examples. The wikipedia article on uniform convergence states:

Some historians claim that Augustin Louis Cauchy in 1921 published a false statement, but with a purported proof, that the pointwise limit of a sequence of continuous functions is always continuous; however, Lakotos offers a re-assessment of Cauchy’s approach. Niels Henrik Abel in 1826 found purported counterexamples to this statement in the context of Fourier series, arguing that Cauchy’s proof had to be incorrect. Cauchy ultimately responded in 1853 with a clarification of his 1821 formulation.

Here on HSM, the limit interpretation is also assumed in the question Why did mathematicians not see that $f_n(x) = x^n$ is a counter-example to Cauchy's "theorem" about limits of continuous functions, which begins with the sentence “In 1821 Cauchy claimed that the limit of a sequence of continuous functions is continuous.”

Regarding the series interpretation, see, for example, the HSM user MJD’s blog posting Cauchy and the continuum

The confusion about Cauchy’s controversial theorem arises from a perennially confusing piece of mathematical terminology: a convergent sequence is not at all the same as a convergent series. Cauchy claimed that a convergent series of continuous functions had a continuous limit. He did not ever claim that a convergent sequence of continuous functions had a continuous limit. But I have often encountered claims that he did that, even though such claims are extremely implausible.

Cauchy’s words, according to a recent translation

I believe that the theorem in question is given as Theorem 1 on page 90 of Bradley and Sandifer’s Cauchy’s Cours d’analyse An Annotated Translation, Springer 2009 :

THEOREM 1 - When the various terms of series (1) are functions of the same variable x, continuous with respect to this variable in the neighborhood of a particular value for which the series converges, the sum s of the series is also a continuous function of x in the neighborhood of this particular value.

As the note (1) indicates, it is important to consider exactly what Cauchy meant by series and convergent series. According to the Bradley and Sandifer's translation, Cauchy uses the following definitions:

We call a series an indefinite sequence of quantities, $$u_0, u_1, u_2, u_3, \dots ,$$ which follow from one to another according to a determined law. These quantities themselves are the various terms of the series under consideration. Let $$s_n = u_0 + u_1 + u_2 + \dots + u_{n-1}$$ be the sum of the first n terms, where n can be any integer number. If, for ever increasing values of n, the sum $s_n$ indefinitely approaches a certain limit s, the series is said to be convergent and the limit in question is called the sum of the series.

So Cauchy appears to be using the term series to describe what we now call a sequence. However, when it comes to a convergent series, Cauchy is not talking about what we now call a convergent sequence. Rather he is using the term in the modern sense of a convergent series - albeit with a different underlying notion of function and the continuum.

This suggests that our "series" interpretation is the correct interpretation. However, as noted above, it is not clear that this translation is stating the original 1821 theorem or the 1853 clarification.

• Nice question. Note that Bradley and Sandifer translated Cauchy's Cours d'Analyse and therefore certainly are dealing with the original 1821 theorem (rather than an 1853 update). Jan 2 '18 at 16:49
• @MikhailKatz Thanks for clarifying that point. It wasn't clear to me at all and searching through google books limited presentation makes it difficult to locate all of the necessary information.
– nwr
Jan 2 '18 at 18:12

Our 2018 publication "Cauchy's infinitesimals, his sum theorem, and foundational paradigms" in Foundations of Science (currently in online first status) deals with this issue in detail. Cauchy's original formulation of the "sum theorem" in 1821 was ambiguous but his 1853 clarification, while also somewhat cryptic, admits a better interpretation in the context of a modern theory of infinitesimals and an enriched continuum than in the context of an Archimedean continuum. Briefly, Cauchy required the condition of convergence also at infinitesimal and other points. Such a pointwise condition for uniform convergence is indeed available in a hyperreal setting; the publication mentioned above provides the appropriate theorem from Abraham Robinson's book.

There is a lot of resistance in the community to accepting the fact that Cauchy's theorem cannot be called "erroneous" (at most "ambiguous") due to defective work of some historians (as evidenced also by the alternative answer to this question); see this MO post for a lively discussion

• The interpretation grounded in non-standard analysis I assume is the re-assessment by Lakotos mentioned in the wiki article I referenced. Thanks for the reference. It is interesting that this is still and active topic of interest to readers - as witnessed by the 2018 publication date.
– nwr
Jan 2 '18 at 16:19
• Thanks. It is not even clear the article will be included in a 2018 issue since there is a big backlog at Foundations of Science. It is most likely to be a 2019 article :-) Jan 2 '18 at 16:41

The correct statement is that is that uniform limit of continuous functions is continuous. Cauchy's proof proofs exactly this. The problem is that uniform convergence was not formalized at the time of Cauchy, so he confused pointwise convergence with uniform convergence.

Remark. Neither was the notion of continuous function formalized. For example, Fourier, a contemporary of Cauchy, defined a continuous function as a function whose graph can be drawn without detaching your pencil from the paper, and according to his definition the "function" $f(x)=1, x>0, f(x)=-1, x<0$ with the graph completed by a vertical segment at $0$ was continuous. He obtained this function as a limit of Fourier series, and he apparently believed Cauchy's theorem. The modern notion of function was formalized by Dirichlet.

• When you find fault the state of formalisation at the time of Cauchy, are you critical of the procedures of masters like Cauchy or rather of the set-theoretic ontology of the entities they used? Jan 2 '18 at 16:48
• I am not sure what Fourier's confusion about continuity has to do with this question. You seem to be trying to pin Fourier's shortcomings on Cauchy. Cauchy was the first author to define continuity as we know it today (as well as Bolzano). Jan 3 '18 at 10:02
• @Mikhail Katz: The question is answered in the first paragraph of my answer. The rest is called "Remark". Jan 3 '18 at 14:11
• The first paragraph of your answer does not address the difference between the two versions of Cauchy's theorem that the OP was specifically asking about: the 1821 version and the 1853 version. Therefore your first paragraph could not in principle be correct, since it implies that there is only one version of the theorem. Also, your use of the term "formalized" is ambiguous, as I already pointed out above. Jan 3 '18 at 14:20