# convergence to level set

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.

• 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. – Invisible May 28 at 17:46