A Banach–Zarecki Theorem for a Banach space-valued function F:[0,1]→X\documentclass[12pt]{minimal}
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\begin{document}$$F : [0,1] \rightarrow X$$\end{document} with compact range is presented. We define the strong absolute continuity (sAC||.||F\documentclass[12pt]{minimal}
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\begin{document}$$sAC_{||.||_{F}}$$\end{document}) and the bounded variation (BV||.||F\documentclass[12pt]{minimal}
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\begin{document}$$BV_{||.||_{F}}$$\end{document}) of F\documentclass[12pt]{minimal}
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\begin{document}$$F$$\end{document} with respect to the Minkowski functional ||.||F\documentclass[12pt]{minimal}
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\begin{document}$$||.||_{F}$$\end{document} associated to the closed absolutely convex hull CF\documentclass[12pt]{minimal}
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\begin{document}$$C_{F}$$\end{document} of F([0,1])\documentclass[12pt]{minimal}
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\begin{document}$$F([0,1])$$\end{document}. It is proved that F\documentclass[12pt]{minimal}
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\begin{document}$$F$$\end{document} is sAC||.||F\documentclass[12pt]{minimal}
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\begin{document}$$sAC_{||.||_{F}}$$\end{document} if and only if F\documentclass[12pt]{minimal}
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\begin{document}$$F$$\end{document} is BV||.||F\documentclass[12pt]{minimal}
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\begin{document}$$BV_{||.||_{F}}$$\end{document}, weak continuous on [0,1]\documentclass[12pt]{minimal}
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\begin{document}$$[0,1]$$\end{document} and satisfies the weak property (N)\documentclass[12pt]{minimal}
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\begin{document}$$(N)$$\end{document}.
