Let M, N be metric spaces with metrics dm and dn respectively.
Let f be a function from M to N.
Suppose that if (an) is any convergent sequence in M, then (f(an)) is a convergent sequence in N.
Show that f is a continuous function.
Note that the question does not say that (an) --> a implies (f(an)) --> f(a). If it did then I could solve the question. But, as it is currently stated, I do not know the answer.