Say we have a function $f:\Bbb R^N\to\Bbb R$ and a scalar $J$, we define $X := \{x | f(x) = J \}$ and assume $X\ne\emptyset$. We have a sequence $\{x_n\}$ s. t. $\{J(x_n)\}\to J$. Under what circumstance we can show $\lim\limits_{n\to\infty} d(x_n, X) = 0$? Here $d(x_n, X) \doteq \inf_{x \in X} \|x - x_n\|$ and $f$ is not bijective.

If $f$ is invertible and $f^{-1}$ is continuous, it would be trivial. So I feel maybe I need to assume something like $f$ is locally invertible, but I don't know how to proceed.

  • $\begingroup$ Use: \Bbb R^N\to\Bbb R and \lim_{n\to\infty} and \|x-x_n\| for the output $\Bbb R^N\to\Bbb R$ and $\lim_{n\to\infty}$ and $\|x-x_n\|$ respectively. $\endgroup$ – Invisible May 28 at 17:46

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