We prove the existence of an additive semigroup of cardinality 2 c contained in the intersection of the classes of Hamel functions (HF) and Sierpi«ski-Zygmund functions (SZ). In addition, we show that under certain set-theoretic assumptions the lineability of the class of Sierpi«ski-Zygmund functions (SZ) is equal to the lineability of the class of almost continuous Sierpi«ski-Zygmund functions (AC ∩ SZ). 1. Introduction The symbols N, Q, and R denote the sets of positive integers, rational and real numbers, respectively. The cardinality of a set X is denoted by the symbol |X|. In particular, |N| is denoted by ω and |R| is denoted by c. We consider only real-valued functions. No distinction is made between a function and its graph. For any two partial real functions f, g we write f + g, f − g for the sum and dierence functions dened on dom(f) ∩ dom(g). We write f |A for the restriction of f to the set A ⊆ R. For any subset Y of a vector space V over the eld E, any v ∈ V , and any e ∈ E we dene v + Y = {v + y : y ∈ Y } and eY = {ey : y ∈ Y }. Recently, there have been lots of attention devoted to nding "large" structures (e.g., vector spaces, algebras) contained in various families of real functions (see [1, 36, 810, 12, 16, 18]). In this article we also consider less restrictive structures like groups and even semigroups. In case of many classes of functions the problem is trivially solved by using already known results about vector spaces contained in those classes (as these vector spaces have maximal possible dimensions). However, in certain situations looking for the "largest" group or semigroup may be of interest. We will recall here some of the most recent denitions related to the theory of lineability
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.