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It is known that two commuting continuous functions on an interval need not have a common fixed point. However, it is not known if such two functions have a common periodic point. we had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic points. In this paper, we first prove that S is a nowhere dense subset of [0,1] if and only if {fÃ¢ÂˆÂˆC([0,1]):Fm(f)Ã¢ÂˆÂ©SÃ‚Â¯Ã¢Â‰Â Ã¢ÂˆÂ…} is a nowhere dense subset of C([0,1]). We also give some results about the common fixed, periodic, and recurrent points of functions. We consider the class of functions f with continuous ÃÂ‰f studied by Bruckner and Ceder and show that the set of recurrent points of such functions are closed intervals

On common fixed points, periodic points, and recurrent points of continuous functions