On strongly countably continuous functions

A real-valued function f on R is strongly countably continuous provided that there is a sequence of continuous functions (fn)n∈N such that the graph of f is contained in the union of the graphs of fn.

Some examples of interesting strongly countably continuous functions are given: one for which the inverse function is not strongly countably continuous, another which is an additive discontinuous function with a big image and a function which is approximately and I-approximately continuous, but it is not strongly countably continuous.