Let X be a compact subset of the complex plane with the property that every relatively open subset of X has positive area and let A0(X)\documentclass[12pt]{minimal}
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\begin{document}$$A_0(X)$$\end{document} denote the space of VMO functions that are analytic on X. A0(X)\documentclass[12pt]{minimal}
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\begin{document}$$A_0(X)$$\end{document} is said to admit a bounded point derivation of order t at a point x0∈∂X\documentclass[12pt]{minimal}
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\begin{document}$$x_0 \in \partial X$$\end{document} if there exists a constant C such that |f(t)(x0)|≤C‖f‖BMO\documentclass[12pt]{minimal}
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\begin{document}$$|f^{(t)}(x_0)|\le C \Vert f\Vert _{{\text {BMO}}}$$\end{document} for all functions in VMO(X)\documentclass[12pt]{minimal}
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\begin{document}$${\text {VMO}}(X)$$\end{document} that are analytic on X. In this paper, we give necessary and sufficient conditions in terms of lower 1-dimensional Hausdorff content for A0(X)\documentclass[12pt]{minimal}
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\begin{document}$$A_0(X)$$\end{document} to admit a bounded point derivation at x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document}. These conditions are similar to conditions for the existence of bounded point derivations on other functions spaces.
