Some remarks about two definitions of continuity

We formulate some variant of the axiom of choice which is neccessary and sufficient for the equivalence ofthe two definitions of continuity (the Cauchy and the Heine definition) for functions from the real line into any metric space. This result definitely solves the problem of the equivalence of these two classical definitions. We also slightly improve one of Sierpiński's results about global continuity of functions from the real line. We give a negative answer to the problem from Jaegermann's ciassical paper [2].