Please check me carefully, but the following approach might help:

We may assume (I hope) that the diameter of the space X is smaller than 1.

Let (x_n) be a dense subset of X. For each n, denote by b_n(x) the first n binary digits of the real number x, (for x in (0,1) ).

The map x\mapsto (d(x,x_n) : n\in N) is continuous and injective (please check, I did not verify). Concatenating the elements of the sequence (b_n(d(x,x_n)) : n), it seems that we obtain an injective continuous image in the Cantor space.

Could it be, thus, that every seperable metrizable space has a continuous bijective image in the Cantor space?

Boaz, any continuous image of connected subset is connected set.

Dear Boaz,

Thanks !

Your solution really answers the question. So we have a continuous bijection from X into  [0,1]  !

It occurred to me that there may be a technical problem with the definition of "the first binary digits", since some numbers have two representation. I hope this is not a real issue, i.e. it can be avoided this way or another, at least for your original question.

l'ensemble2^N  a la puissance du continu

donc l'applicatio F= 2 puissance de R dans R vérifie

F(Q) non dénombrable

Gilbert, could you provide more details, and in English. I wish I knew French, but unfortunately I don't, and google translate helps little with math texts. :(

Unfortunately my translator is not very clear translates from French.

Could you please explain in more detail.

Excuse me I have not exactly understood your Answer

Chèr Gilbert que  voulez- vous dire par l'ensemble 2N ? it-il l'ensemble des parties de N?

If Gilbert meant by 2N the set of all subsets of Natural numbers N, then he meant that 2N is equipotent to the set of real numbers R.  So the map f which x---->2x from R to R satisfies f(Q)= uncountable set, where Q is the set of rationals.

Also what provided Boaz in his solution is close to what said Gilbert, you can see this link, section 2, but it is in french: 

https://fr.wikipedia.org/wiki/Puissance_du_continu

Hey, you can use key word: equipotency