We prove that the set of Lipschitz functions with the property of having a maximal Clarke subdifferential at every point contains a linear subspace of uncountable dimension. Our approach is constructive. In particular, in strong contrast to a previous result of similar flavour, by J. Borwein and X. Wang, we do not make use of the Baire category theorem.
In particular we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions. (Joint work with G. Flores, University of Chile)