Graphs of real functions with pathological behaviors

This paper is dedicated to the memory of Franco Montagna. I present and discuss functions from R to R whose graphs have unexpected properties. For instance: functions which are periodic and, at the same time, increasing quasiperiodic and decreasing quasiperiodic, functions whose graphs are dense in the plane but in some sense behave like straight lines, everywhere surjective functions (or strong Darboux function). In this context some theorems are stated.