Independent Bernstein sets and algebraic constructions

We present the method of constructing algebras and linear spaces of 2 c generators using independent Bernstein sets. As an application we obtain large algebras of special functions: (i) strongly everywhere surjective functions which are not perfectly everywhere surjective; (ii) nowhere continuous Darboux functions; (iii) nowhere continuous compact to compact functions; (iv) functions which are continuous precisely on a fixed closed proper subset of R . Most conclusions obtained in this paper are improvements of some already known results.