Suppose $f(a) = f(b)$ and let $(x_n)_n = (a,b,a,b, \ldots)$. $(f(x_n))_n$ is constant, hence convergent, so $(x_n)_n$ must converge. Then $a=b$ and $f$ is injective. But then $f^{-1}:\mathbb{R} \to \mathbb{R}$ is a continuous bijection and it is a well known that this implies that $f^{-1}$ is an homeomorphism, so $f$ is continuous.