NON-LIPSCHITZ DIFFERENTIABLE FUNCTIONS ON SLIT DOMAINS L. BERNAL-GONZ ´ ALEZ * , P. JIM ´ ENEZ-RODR ´ IGUEZ ** , G.A. MU ˜ NOZ-FERN ´ ANDEZ ** , AND J.B. SEOANE-SEP ´ ULVEDA ** Abstract. It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside the family of non-Lipschitz differentiable real functions with boun- ded gradient defined on special non-convex plane domains. In particular, this yields that there are many differentiable functions on plane domains that do not satisfy the Mean Value Theorem. 1. Introduction A standard result from Real Analysis states that, for any interval I of the real line R, a differentiable function f : I −→ R is Lipschitz if and only if it has bounded derivative. Recall that a mapping ϕ :(X,d 1 ) → (Y,d 2 ) between two metric spaces is called Lipschitz whenever there is a constant k ∈ (0, +∞) such that d 2 (ϕ(P ),ϕ(Q)) ≤ kd 1 (P,Q) for all P,Q ∈ X . One could think if the above mentioned result still holds for domains of sev- eral variables. If N ∈ N := {1, 2, 3,... }, by a domain of R N we mean a nonempty connected open subset Ω ⊂ R N . If Ω is a convex domain, a sim- ple application of the Mean Value Theorem yields that every differentiable function f :Ω → R with bounded gradient is Lipschitz. This statement does not hold if Ω is not convex. In fact, Jim´ enez, Mu˜ noz and Seoane [11, Theorem 3.2] showed in 2012 that even in the case of the (very simple, nonconvex) domain of R 2 given by D = {(x,y) ∈ R 2 : x 2 + y 2 < 1}\{(x,y) ∈ R 2 : x = 0 and y ≥ 0} there is a plethora of such non-Lipschitz functions. Their result reads as follows. By c we denote the cardinality of the continuum. Theorem 1.1. The set of differentiable functions f : D → R with bounded gradient, non-Lipschitz and therefore not satisfying the classical Mean Value Theorem contains, except for the zero function, a c-dimensional vector space. 2010 Mathematics Subject Classification. Primary 26B35; Secondary 15A03, 31C05. Key words and phrases. Non-Lipschitz function, differentiable function, domain in the plane, free algebra. * Supported by the Plan Andaluz de Investigacin de la Junta de Andaluca FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. ** Supported by the Spanish Ministry of Science and Innovation, Grant MTM2012- 34341. 1