# Smooth approximation

How one can show, that if $f(x_1,\ldots,x_n)$ is a continuous function on an open subset $U\subset \mathbb{R}^n$, then for every $\varepsilon > 0$ and every open $V\subset U$, such that $\bar V \subset U$, there exists a function $g(x_1,\ldots,x_n)$, such that:

1) $g$ is smooth on $V$;

2) $g|_{U-V}=f|_{U-V}$;

3) $\max_{x \in \bar V} |f(x)-g(x)|\leq \varepsilon$;

4) $g$ is smooth in all points, where $f$ is smooth.

This lemma is used in the book of Dubrovin, Novikov, Fomenko, but they don't prove it and don't give some refernce, where it can be found.

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