Cauchy proved a sum theorem for series of continuous functions in 1821, and published another article on the subject in 1853.

Michael Segre, writing in Archive for History of Exact Sciences, claimed concerning Cauchy's sum theorem:

What is amazing here is Cauchy's attitude. He totally disregarded Fourier's counterexample and did not admit having made a mistake: not only did he "prove" his theorem, but he repeated it in a paper read to the Academie des Sciences as late as 1853. (page 233 in Segre, Michael. Peano's axioms in their historical context. Arch. Hist. Exact Sci. 48 (1994), no. 3-4, 201-342)

For his part, Umberto Bottazzini wrote:

The language of infinites and infinitesimals that Cauchy used here seemed ever more inadequate to treat the sophisticated and complex questions then being posed by analysis... The problems posed by the study of nature, such as those Fourier had faced, now reappeared everywhere in the most delicate questions of "pure" analysis and necessarily led to the elaboration of techniques of inquiry considerably more refined than those that had served French mathematicians at the beginning of the century. Infinitesimals were to disappear from mathematical practice in the face of Weierstrass' epsilon and delta notation (p. 208 in Bottazzini, Umberto. The higher calculus: a history of real and complex analysis from Euler to Weierstrass. Translated from the Italian by Warren Van Egmond. Springer-Verlag, New York, 1986)

These authors make Cauchy appear rather obstinate with regard to what is described by some historians as his famous "mistake". To those mathematicians who have studied Cauchy's work, such claims by historians seem surprising. Are we to accept them at face value? Is there more to the story than meets the eye?

The analysis by my coauthors and myself is presented in this article.

Note 1. I found the article by Cleave mentioned in Carlo's answer somewhat disappointing. This is because Cleave overplays his hand by making an excessive claim for Cauchy by seeking to attribute the notion of uniform convergence to Cauchy as early as 1821. Cleave thus undermines what could have otherwise been a strong case for Cauchy's prescience that can be detected in the 1853 article that details the required condition (unlike the 1821 book). I don't think a strong claim can be made concerning such prescience by Cauchy in 1821 because the 1821 statement is just too ambiguous to attach that much significance to it. Also, Cleave's case is undermined by the fact that Cauchy seems to be clearly stating in 1853 that he is modifying the 1821 hypothesis, by including a requirement of convergence also at infinitesimal points, etc. A much better case was made by Detlef Laugwitz in his articles published in the 1980s.

Note 2. This question is within the scope of the typical questions posed here under the ho.history-overview tag.