A note on uniform convergence and transitivity

In this paper, let (X, d) be a metric space. Let fn: X → X be a sequence of continuous and topologically transitive functions such that (fn) converges uniformly to a function f. It is shown that if (X, d  ) is compact and perfect, limn→∞d∞fnn,fn=0 and fnn(x) is dense in X for some x ∈ X, then f is totally transitive. We also present a sufficient condition for f to be topologically transitive (resp. syndetically transitive). Furthermore, we give a sufficient condition for f to be topologically weak mixing (resp. topologically mixing).In addition, for a compact metric space (X, d), suppose that the fn: X → X are continuous and converge uniformly to f. If for a given ε > 0, there exists a positive integer n0 such that for all n > n0 and all l > 0, d(fnl(x),fl(x))<ε for all x ∈ X, then the following statements hold:(1)f is syndetically sensitive (resp. cofinitely sensitive) whenever the fn are syndetically sensitive (resp. cofinitely sensitive).(2)f is multi-sensitive whenever the fn are multi-sensitive.(3)If f is sensitive (resp. cofinitely sensitive) with δ as a constant of sensitivity, then there exists an integer N > 0 such that fn is sensitive (resp. cofinitely sensitive) with 13δ as a constant of sensitivity for any n ⩾ N.(4)If f is multi-sensitive (resp. syndetically sensitive) with δ as a constant of sensitivity, then there exists an integer N > 0 such that fn is multi-sensitive (resp. syndetically sensitive) with 19δ as a constant of sensitivity for any n ⩾ N.