If you replace the reals $\mathbb{R}$ with Cantor space $2^\omega$ or with Baire space $\omega^\omega$ (homeomorphic to the space of irrationals), then the answer is yes. Indeed, one can have the function defined on the whole space of $\omega$-sequences, not just the injective ones.
To see this, define $f(x_0,x_1,...)=(y_0,y_1,...)$, where $y_k$ extends the $k^{th}$ finite sequence $u_k$, and diagonalizes the $x_n's$ in a canonical way beyond the length of $u_k$, so that the $|u_k|+j^{th}$ digit of $y_k$ is different from the $|u_k|+j^{th}$ digit of $x_j$. This is a continuous function, since any finitely many digits for the output are determined by finitely many digits of the input. The $y_k$'s are not among the $x_n$, since they diagonalize against this list, and the $y_k$'s are dense, because $y_k$ extends $u_k$. (By changing the diagonalization procedure slightly, it is easy to arrange that the $y_k$ are all distinct.)