Discontinuous, although “highly” differentiable, real functions and algebraic genericity

Abstract

We exhibit a class of functions f:R→R which are bounded, continuous on R∖Q, left discontinuous on Q, right differentiable on Q, and upper left Dini differentiable on R∖Q. Other properties of these functions, such as jump sizes and local extrema, are also discussed. These functions are constructed using probabilistic methods. We also show that the families of functions satisfying similar properties contain large algebraic structures (obtaining lineability, algebrability and coneability).