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We show that the directed subdifferential introduced for differences ofconvex (delta-convex, DC) functions by Baier and Farkhi can be constructed fromthe directional derivative without using any information on the DC structure ofthe function. The new definition extends to a more general class of functions,which includes Lipschitz functions definable on o-minimal structure andquasidifferentiable functions.

Directed Subdifferentiable Functions and the Directed Subdifferential without Delta-Convex Structure