Let $X$ be a topological space, and let $f, g: S^1\to X$ be homotopic circles in $X$. Is there a continuous function from $X$ to itself that sends $f(S^1)$ to $g(S^1)$, and if so, how do I find it?
This is related to a homework problem, so please only give hints.
Hint: There are easy counterexamples. Just think about how different the image sets $f(S^1)$ and $g(S^1)$ can be.