  In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his
1821 book Cours d'Analyse, may have plagiarized Bolzano's book 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
K\"astner (in Section 183 of his 1766 book), which the former seems to have
misrepresented at least partially.
  Cauchy's treatment of continuity goes back at least to his 1817 course
summaries, refuting a key component of Grattan-Guinness' plagiarism hypothesis
(that Cauchy may have lifted continuity from RB after reading it in a Paris
library in 1818). We explore antecedents of Cauchy and Bolzano continuity in
the writings of K\"astner and earlier authors.
