Continuity between Cauchy and Bolzano: issues of antecedents and priority

In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his 1821 Cours d'Analyse, may have plagiarized Bolzano's Rein analytischer Beweis (RB), first published in 1817. That paper was subsequently discredited in several works, but some of its assumptions still prevail today. In particular, it is usually considered that Cauchy did not develop his notion of the continuity of a function before Bolzano developed his in RB and that both notions are essentially the same. We argue that both assumptions are incorrect, and that it is implausible that Cauchy's initial insight into that notion, which eventually evolved to an approach using infinitesimals, could have been borrowed from Bolzano's work. Furthermore, we account for Bolzano's interest in that notion and focus on his discussion of a definition by Kästner (in Section 183 of his 1766 book), which the former seems to have misrepresented at least partially.