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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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migrated from mathoverflow.net Jul 16 '13 at 22:36

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The proof can be found in several places. For instance, see Thm 2.5 in Hirsch, M. W., Differential Topology (Springer-Verlag, 1976) or §6.7 in Steenrod, N., The Topology of Fibre Bundles (PUP, 1951). Unfortunately, these sources don't provide a name or a historical note about this result.

A recent paper, where I looked up these references is Wockel, C., A Generalisation of Steenrod's Approximation Theorem (2006, arXiv, dmlcz), which supplies an obvious name.

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