New Zealand is not on the same continental shelf and so is not part of the continent of Australia but is part of the submerged continent Zealandia. Zealandia and Australia together are part of the wider region known as Oceania or Australasia.
Just so we're clear :)
The usual argument people falsely cling to is:
As a cultural construct, the concept of a continent may go beyond the continental shelf to include oceanic islands and continental fragments. In this way, Iceland is considered part of Europe and Madagascar part of Africa. Extrapolating the concept to its extreme, some geographers take Australia, New Zealand and all the islands of Oceania (or sometimes Australasia) to be equivalent to a continent, allowing the entire land surface of the Earth to be divided into continents or quasi-continents.