Uniform convergence and transitivity

Let (X, d) be a metric space and fn : X → X a sequence of continuous functions such that (fn) converges uniformly to a function f. If fn is transitive for all n∈Nn∈N, then the purpose of this work is, on the one hand, to show that f is not necessarily transitive and, on the other, to give sufficient conditions for the transitivity of the limit function f.