Let K and S be locally compact Hausdorff spaces and X a Banach space. Suppose that T is a linear operator from C0(K)\documentclass[12pt]{minimal}
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\begin{document}$$C_{0}(K)$$\end{document} into C0(S,X)\documentclass[12pt]{minimal}
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\begin{document}$$C_{0}(S, X)$$\end{document} with ‖T‖‖T-1‖<λ(X),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert T\Vert \ \Vert T^{-1}\Vert < \lambda (X), \end{aligned}$$\end{document}where λ(X)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (X)$$\end{document} is a parameter introduced by Jarosz in 1989. We prove that there exist a subset S0\documentclass[12pt]{minimal}
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\begin{document}$$S_0$$\end{document} of S and a continuous function from S0\documentclass[12pt]{minimal}
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\begin{document}$$S_0$$\end{document} onto K. This vector-valued version of the 1966 classical Holsztyński’s theorem is optimal in the case where X=lp\documentclass[12pt]{minimal}
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\begin{document}$$X=l_{p}$$\end{document}, 2≤p<∞\documentclass[12pt]{minimal}
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\begin{document}$$ 2 \le p< \infty $$\end{document}. Moreover, if T satisfies the following stronger condition ‖T‖‖T-1‖<3λ(X)λ(X)+2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert T\Vert \ \Vert T^{-1}\Vert < \frac{3\lambda (X)}{\lambda (X)+2}, \end{aligned}$$\end{document}then for each ordinal α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} there exist a subset Sα\documentclass[12pt]{minimal}
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\begin{document}$$S_{\alpha }$$\end{document} of the α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}th derivative of S and a continuous function from Sα\documentclass[12pt]{minimal}
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\begin{document}$$S_{\alpha }$$\end{document} onto the α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}th derivative of K. When K is compact, the set Sα\documentclass[12pt]{minimal}
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\begin{document}$$S_{\alpha }$$\end{document} may be taken closed.
