Every K-Lipschitz function from a closed subset of a metric space into any real interval extends to a K-Lipschitz function on the whole space, and analogous extension, decomposition, and selection results hold for locally Lipschitz functions.
Aggarwal, Split Lipschitz-type functions , Topology Appl
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On real-valued functions of Lipschitz type
Every K-Lipschitz function from a closed subset of a metric space into any real interval extends to a K-Lipschitz function on the whole space, and analogous extension, decomposition, and selection results hold for locally Lipschitz functions.