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\begin{document}$$\epsilon >0$$\end{document}, n a non-negative integer, and A a complex unital Banach algebra. Define γn:A×C→[0,∞]\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _n: A\times {\mathbb {C}}\rightarrow [0,\infty ]$$\end{document} by γn(a,z)=‖(z-a)-2n‖-1/2n,if(z-a)isinvertible0,if(z-a)is not invertible.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \gamma _n(a,z)={\left\{ \begin{array}{ll} \Vert (z -a)^{-2^n}\Vert ^{-1/2^n}, &{}\quad \text {if } (z-a) \text{ is } \text{ invertible }\\ 0, &{}\quad \text {if } (z-a) \text { is not invertible}. \end{array}\right. } \end{aligned}$$\end{document}The (n,ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$(n,\epsilon )$$\end{document}-pseudospectrum Λn,ϵ(a)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda _{n,\epsilon }(a)$$\end{document} of an element a∈A\documentclass[12pt]{minimal}
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\begin{document}$$a\in A$$\end{document} is defined by Λn,ϵ(a):={λ∈C:γn(a,λ)≤ϵ}\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda _{n,\epsilon }(a):= \{\lambda \in {\mathbb {C}}:\gamma _n(a,\lambda )\le \epsilon \}$$\end{document}. We show that γ0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _0$$\end{document} is Lipschitz on A×C\documentclass[12pt]{minimal}
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\begin{document}$$A\times {\mathbb {C}}$$\end{document}, γn\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _n$$\end{document} is uniformly continuous on bounded subsets of A×C\documentclass[12pt]{minimal}
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\begin{document}$$A\times {\mathbb {C}}$$\end{document} for n≥1\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 1$$\end{document}, and γn\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _n$$\end{document} is Lipschitz on some particular domains for n≥1\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 1$$\end{document}. Using these properties, we establish that the map (ϵ,a)↦Λn,ϵ(a)\documentclass[12pt]{minimal}
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\begin{document}$$(\epsilon ,a)\mapsto \Lambda _{n,\epsilon }(a)$$\end{document} is continuous at (ϵ0,a0)\documentclass[12pt]{minimal}
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\begin{document}$$(\epsilon _0,a_0)$$\end{document} if and only if the level set {λ∈C:γn(a0,λ)=ϵ0}\documentclass[12pt]{minimal}
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\begin{document}$$\{\lambda \in {\mathbb {C}}: \gamma _n(a_0,\lambda )= \epsilon _0 \}$$\end{document} does not contain any non-empty open set. In particular, this happens when a is a compact operator on a Banach space or a bounded operator on a Hilbert space or on an Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p $$\end{document} space with 1≤p≤∞\documentclass[12pt]{minimal}
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\begin{document}$$1\le p\le \infty $$\end{document}. We also give examples of operators where this condition is not satisfied, and consequently, the map is not continuous.
