Periodic cycles for an extension of generalized 3x + 1 functions

The Collatz conjecture is an open problem involving the 3x+1 function. The function belongs to
a class of generalized 3x + 1 functions of relatively prime type. This paper focuses on exploring periodic cycles
for an extension of a generalized 3x + 1 function of relatively prime type. By extending its domain to R, the
result shows that every integer periodic point is isolated in the usual topology on R. Moreover, every positive
integer periodic cycle for the extension is attracting if the generalized 3x + 1 function is satisfied by parameters
under some conditions.