Lineability, Spaceability, and Additivity Cardinals for Darboux-like Functions

We introduce the concept of maximal lineability cardinal number, mL(M), of a subset M of a topological vector space and study its relation to the cardinal numbers known as: additivity A(M), homogeneous lineability HL(M), and lineability L(M) of M . In particular, we will describe, in terms of L, the lineability and spaceability of the families of the following Darboux-like functions on Rn, n ≥ 1: extendable, Jones, and almost continuous functions. 1. Preliminaries and background The work presented here is a contribution to a recent ongoing research concerning the following general question: For an arbitrary subset M of a vector space W , how big can be a vector subspace V contained in M∪{0}? The current state of knowledge concerning this problem is described in the very recent survey article [4]. So far, the term big in the question was understood as a cardinality of a basis of V ; however, some other measures of bigness (i.e., in a category sense) can also be considered. Following [1,23] (see, also, [13]), given a cardinal number μ we say that M ⊂ W is μ-lineable if M ∪{0} contains a vector subspace V of the dimension dim(V ) = μ. Consider the following lineability cardinal number (see [2]): L(M) = min{κ : M ∪ {0} contains no vector space of dimension κ}. Notice that M ⊂ W is μ-lineable if, and only if, μ < L(M). In particular, μ is the maximal dimension of a subspace of M ∪ {0} if, and only if, L(M) = μ. The number L(M) need not be a cardinal successor (see, e.g., [1]); thus, the maximal dimension of a subspace of M ∪ {0} does not necessarily exist. If W is a vector space over the field K and M ⊂ W , let st(M) = {w ∈ W : (K \ {0})w ⊂ M}.