Cauchy published a "proof" that a convergent sequence of continuous functions converges to a continuous function, relying on a not-completely-rigorous idea of continuity. This is particularly notable given Cauchy's role in giving precise definitions here, and also given how easy it is to think of counterexamples. 

The discussion in the following link may be relevant:
http://www.math.usma.edu/people/Rickey/hm/CalcNotes/CauchyWrgPr.pdf