A new class of functions called “Rz-supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of Rz-supercontinuous functions properly includes the class of Rcl-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of cl-supercontinuous (≡ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983), and is strictly contained in the class of Rδ-supercontinuous, Kohli, Tyagi, Singh, Aggarwal (2014), which in its turn is properly contained in the class of R-supercontinuous functions, Kohli, Singh, Aggarwal (2010).