Continuous selections for the metric projection onC 1

C ₁(K) is the space of real continuous functions onK endowed with the usualL ₁,-norm where \(K = \overline {\operatorname{int} K}\) is compact inR ^{m · U} is a finite-dimensional subspace ofC ₁,(K). The metric projection ofC ₁,(K) ontoU contains a continuous selection with respect toL ₁, -convergence if and only ifU is a unicity (Chebyshev) space forC ₁,(K). Furthermore, ifK is connected andU is not a unicity space forC ₁,(K), then there is no continuous selection with respect toL _∞-convergence. An example is given of aU and a disconnectedK with no continuous selection with respect toL ₁-convergence, but many continuous selections with respect toL _∞-convergence.