The finite time disentanglement or entanglement sudden death, when only one part of the composite system is subjected to a single noise, is examined. While it is shown that entanglement sudden death canoccur when a part of the entangled mixed state is subjected to either amplitude noise or phase noise,local action of either of them does not cause entanglement sudden death in pure entangled states. In contrast, depolarizing noise is shown to have an ability to cause sudden death of entanglement even in pureentangled states, when only one part of the state is exposed to it. The result is illustrated through theaction of different noisy environments individually on a single qubit of the so-called X class of statesand an arbitrary two-qubit pure state.