Smooth approximation theorem for differential manifolds

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This fact is an application of the following pivotal fact/result/idea: Stone-Weierstrass theorem
View other applications of Stone-Weierstrass theorem OR Read a survey article on applying Stone-Weierstrass theorem

This article gives a result on smooth approximation; a result stating that a continuous map can be replaced by a smooth map satisfying similar properties

Statement

Let M and N be differential manifolds and define C^0(M,N) to be the space of all continuous maps from M to N, endowed with the compact-open topology. Let C^\infty(M,N) be the space of all smooth maps from M to N, viewed as a subset of C^0(M,N). Then C^\infty(M,N) is dense in C^0(M,N).

Applications