Lineability of sets of nowhere analytic functions

Abstract

Although the set of nowhere analytic functions on [ 0 , 1 ] is clearly not a linear space, we show that the family of such functions in the space of C ∞ -smooth functions contains, except for zero, a dense linear submanifold. The result is even obtained for the smaller class of functions having Pringsheim singularities everywhere. Moreover, in spite of the fact that the space of differentiable functions on [ 0 , 1 ] contains no closed infinite-dimensional manifold in C ( [ 0 , 1 ] ) , we prove that the space of real C ∞ -smooth functions on ( 0 , 1 ) does contain such a manifold in C ( ( 0 , 1 ) ) .