Unraveling a tangle

Question

Let $f: [0,1] \to X$ be a continuous function from the segment $[0,1]$ to a Hausdorff space $X$ such that $f(0) \ne f(1)$.

Can we claim that there is always an injective continuous function $g: [0,1] \to X$ such that $g(0)=f(0)$ and $g(1)=f(1)$ ?