= Smooth approximation of maps into Euclidean space
A <continuous map> from a <smooth manifold> to $\mathbb R^d$ can be approximated by a <smooth map> with any prescribed positive continuous pointwise error bound. In coordinate charts, smoothing by local <convolution> and then combining with a smooth <partition of unity> gives the approximation. On a compact subset of an open chart domain, a smooth cutoff allows the resulting straight-line <homotopy> to be supported in that domain. This permits the <Sard theorem> to be used after approximation; the original continuous map can be space-filling and need not omit a point.
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