# Unraveling a tangle [duplicate]

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 $$g(1)=f(1)$$ ?
• Does $g$ need to have any relation to $f$ other than at endpoints $0,1$? – coffeemath May 3 at 4:17