Prove or give counterexample Every sequence of real numbers

Prove or give counterexample: Every sequence of real numbers is a continuous function.

Solution

It is not clear what is to be proved/disproved.

1) If this is a statement concerning a sequence of real numbers being the image of a continuous function on R, then it is definitely false. The image of a connected set being connected, the only such sequence is a constant sequence. In other words the function f must map every x to the same number c, say.

2) On the other hand, if we consider a countable discrete topological space X, then any real sequence could be the image of a continuous function on X. (as any function on X is continuous)

It would be better to repost this problem with the correct statement and notation.