In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. In the discrete-time case, this is true for all continuous maps on metric spaces. In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.
in some caSes, The LasT tWo pRoPeRtieS abOve haVe beeN shOwn to aCtuallY impLY sEnsitiViTy TO initiAl condiTioNs. iN ThE disCrEtE-tiMe case, tHis iS TrUe For All coNTInuouS maPS ON metric SPaCes. in these CasEs, wHile it is oFtEn the mOst pracTIcAlly SIgnificanT prOpeRTy, "SEnsitivity to InitiaL conDitIoNs" need not Be staTED iN ThE definitioN.