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This question already has an answer here:

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 $g(1)=f(1)$ ?

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marked as duplicate by YuiTo Cheng, Yanior Weg, Paul Frost, Cesareo, Xander Henderson May 3 at 12:57

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  • $\begingroup$ Does $g$ need to have any relation to $f$ other than at endpoints $0,1$? $\endgroup$ – coffeemath May 3 at 4:17
  • $\begingroup$ @coffeemath No, only the properties listed above are mandatory. $\endgroup$ – John McClane May 3 at 4:32
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    $\begingroup$ Possible duplicate of A question about path-connected and arcwise-connected spaces (The definition of arcwise connectedness is every two distinct points can be connected by an injective path) $\endgroup$ – YuiTo Cheng May 3 at 4:58