I started working with a map $f:\mathbb{R} \to \mathbb{R}$ such that it is continuous except on a finite set. I started looking for a definition of topological transitivity and topological mixing in this context. So, I am asking if there is a good reference where I can find such definition.

I am aware the works of Kolyada et. al. where some of these aspects are discussed, however the considered maps are always continuous. Also, I know some references where topological transitivity for piecewise continuous maps on the unit interval is discussed, for example Glendinning's work "Topological conjugation of Lorenz maps by β-transformations".

Thanks in advance for your help.