As embodied in the title of the paper strong and weak variants of continuity that lie strictly between strong continuity of Levine and almost continuity due to Singal and Singal are considered. Basic properties of almost completely continuous functions (≡ R-maps) and δ-continuous functions are studied. Direct and inverse transfer of topological properties under almost completely continuous functions and δ-continuous functions are investigated and their place in the hier- archy of variants of continuity that already exist in the literature is out- lined. The class of almost completely continuous functions lies strictly between the class of completely continuous functions studied by Arya and Gupta (Kyungpook Math. J. 14 (1974), 131-143) and δ-continuous functions defined by Noiri (J. Korean Math. Soc. 16, (1980), 161-166). The class of almost completely continuous functions properly contains each of the classes of (1) completely continuous functions, and (2) al- most perfectly continuous (≡ regular set connected) functions defined by Dontchev, Ganster and Reilly (Indian J. Math. 41 (1999), 139-146) and further studied by Singh (Quaestiones Mathematicae 33(2)(2010), 1–11) which in turn include all δ-perfectly continuous functions initi- ated by Kohli and Singh (Demonstratio Math. 42(1), (2009), 221-231) and so include all perfectly continuous functions introduced by Noiri (Indian J. Pure Appl. Math. 15(3) (1984), 241-250).
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机译：正如论文标题所体现的，考虑了严格的Levine强连续性和Singal和Singal导致的几乎连续性之间的强弱连续性变体。研究了几乎完全连续的函数（≡R-映射）和δ-连续函数的基本性质。研究了拓扑性质在几乎完全连续函数和δ连续函数下的直接和逆向传递，并概述了它们在文献中已经存在的连续性变量层次中的位置。几乎完全连续的函数的类别严格介于Arya和Gupta研究的完全连续的函数的类别（Kyungpook Math。J. 14（1974），131-143）和Noiri定义的δ-连续函数（J. Korean Math。 （Soc.16，（1980），161-166）。几乎完全连续的函数类别正确地包含（1）完全连续的函数类别，以及（2）Dontchev，Ganster和Reilly（Indian J. Math。 41（1999），139-146）并由Singh进行了进一步研究（Quaestiones Mathematicae 33（2）（2010），1-11），这又包括Kohli和Singh提出的所有δ完美连续函数（演示数学。 42（1），（2009），221-231）等包含Noiri引入的所有完全连续函数（Indian J. Pure Appl。Math。15（3）（1984），241-250）。
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