A note on Devaney's definition of chaos

Devaney defines a function to be chaotic if it satisfies three conditions: transitivity, having dense set of periodic points and sensitive dependence on initial conditions. Banks et al prove that if the function is continuous then the third conditioin is implied from the first two and therefore is redundant. However, if the function is not assumed to be continuous, then it is not known if the third condition is redundant or not. In this note, without assuming the function is continuous, we prove that the third condition is redundant if the underlying topological space is not precompact.