I started studying topology and encountered the epsilon-delta definition of continuity applied for general metric spaces. From my calculus courses I am used to thinking of both $\epsilon$ and $\delta$ in the epsilon-delta definition of continuity as limits tending to zero. That is (as I understand now) is equivalent to the statement that a continuous function maps every converging Cauchy sequence from its domain into a converging Cauchy sequence in its range.
However what if there are no Cauchy sequences in some metric space (or they just do not converge in this space)? Does it mean that on such domain no continuous functions can be defined?