Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his book that "Weierstrass undoubtedly saw himself as Cauchy's heir in analysis and so helped to create the belief that Cauchy's achievements included ideas that were actually his own" (The development of the foundations of mathematical analysis from Euler to Riemann, page 120). In fact, Cauchy apparently never gave an epsilon-delta definition of either limit or continuity. His infinitesimal definition of continuity ("infinitesimal $x$-increment always produces infinitesimal $y$-increment") is reproduced in a number of secondary studies, for example in J. Gray (Plato's Ghost, page 64). The question is then, who in fact gave you epsilon-delta: Cauchy or Weierstrass? Editors are requested to refrain from answers based purely on opinion, but rather to base themselves on published sources supporting either of the two views. In particular, a primary source in Cauchy giving an epsilon-delta definition of continuity would be welcome.
Note 1. Grabiner's example cited by @Brian illustrates the issue well. Cauchy infrequently uses preliminary forms of epsilon, delta arguments (notice the opening "let delta, epsilon be very small numbers" which would certainly not make it into calculus texts today) in some of his proofs, but he never gave an epsilon, delta definition. This example suggests that Cauchy never gave such a definition, for if he did, Grabiner would have likely cited it. Cauchy's definition of continuity is "infinitesimal $x$-increment always produces an infinitesimal change in $y$". This is closer to the modern infinitesimal definition of continuity than to the modern epsilon, delta definition; in fact it looks identical to the modern hyperreal definition at the syntactic level.
Best Answer
It was Bernard Bolzano. According to Wikipedia, "In calculus, the (ε, δ)-definition of limit ... is a formalization of the notion of limit. It was first given by Bernard Bolzano in 1817, followed by a less precise form by Augustin-Louis Cauchy."