Abstract:
We study the problem of the existence of a continuous selection for the metric projection to the set of $n$-link piecewise-linear functions in the space $C[0,1]$. We show that there is a continuous selection if and only if $n=1$ or $n=2$. We establish that there is a continuous
$\varepsilon$-selection to $L$ ($L\subset C[0,1]$) if $L$ belongs to a certain class of sets that contains, in particular, the set of algebraic rational fractions and the set of piecewise-linear functions. We construct an example showing that sometimes there is no $\varepsilon$-selection
for a set of splines of degree $d>1$.
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