Let ϵ> 0 , n a non-negative integer, and A a complex unital Banach algebra. Define γn: A× C→ [0 , ∞] by γn(a,z)={‖(z-a)-2n‖-1/2n,if(z-a)isinvertible0,if(z-a)is not invertible.The (n, ϵ) -pseudospectrum Λ n , ϵ(a) of an element a∈ A is defined by Λ n , ϵ(a) : = { λ∈ C: γn(a, λ) ≤ ϵ}. We show that γ is Lipschitz on A× C, γn is uniformly continuous on bounded subsets of A× C for n≥ 1 , and γn is Lipschitz on some particular domains for n≥ 1. Using these properties, we establish that the map (ϵ, a) ↦ Λ n , ϵ(a) is continuous at (ϵ, a) if and only if the level set { λ∈ C: γn(a, λ) = ϵ} 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 space with 1 ≤ p≤ ∞. We also give examples of operators where this condition is not satisfied, and consequently, the map is not continuous. © 2019, Springer Nature Switzerland AG.}, author_keywords={(n, ϵ) -Pseudospectrum; Banach algebra; Pseudospectrum; Spectrum}, funding_details={2/39(2)/2015/NBHM/R&D-II/7440}, funding_text_1={The authors would like to thank the anonymous referee for his/her valuable comments that led to a considerable improvement in this paper. The first author would like to thank the Department of Atomic Energy (DAE), India (Ref No: 2/39(2)/2015/NBHM/R&D-II/7440) for financial support.}, references={Bögli, S., Siegl, P., Remarks on the convergence of pseudospectra (2014) Integral Equ. Oper. Theory, 80, pp. 303-321; Böttcher, A., Pseudospectra and singular values of large convolution operators (1994) J. Integral Equ. Appl., 6, pp. 267-301; Böttcher, A., Grudsky, S.M., Silbermann, B., Norms of inverses, spectra, and pseudospectra of large truncated Wiener–Hopf operators and Toeplitz matrices (1997) N. Y. J. Math., 3, pp. 1-31; Böttcher, A., Grudsky, S.M., Can spectral value sets of Toeplitz band matrices jump? (2002) Linear Algebra Appl., 351 (352), pp. 99-116; Davies, E.B., Pseudospectra of differential operators (2000) J. Oper. Theory, 43, pp. 243-262; Davies, E.B., A defence of mathematical pluralism (2005) Philos. Math., 13 (3), pp. 252-276; Dhara, K., Kulkarni, S.H., The (n, ϵ) -pseudospectrum of an element of a Banach algebra (2018) J. Math. Anal. Appl., 464, pp. 939-954; Dunford, N., Schwartz, J.T., (1958) Linear Operators. I. General Theory, , Interscience Publishers Ltd., London; Gallestey, E., Hinrichsen, D., Pritchard, A.J., Spectral value sets of closed linear operators (2000) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456, pp. 1397-1418; Globevnik, J., On complex strict and uniform convexity (1975) Proc. Am. Math. Soc., 47, pp. 175-178; Globevnik, J., Norm-constant analytic functions and equivalent norms (1976) Ill. J. Math., 20, pp. 503-506; Hansen, A.C., On the approximation of spectra of linear operators on Hilbert spaces (2008) J. Funct. Anal., 254, pp. 2092-2126; Hansen, A.C., Infinite-dimensional numerical linear algebra: theory and applications (2010) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466, pp. 3539-3559; Hansen, A.C., On the solvability complexity index, the n -pseudospectrum and approximations of spectra of operators (2011) J. Am. Math. Soc., 24, pp. 81-124; Hansen, A.C., Nevanlinna, O., Complexity issues in computing spectra, pseudospectra and resolvents (2017) Banach Cent. Publ., 112, pp. 171-194; Harrabi, A., Pseudospectre d’une suite d’opérateurs bornés (1998) RAIRO Modél. Math. Anal. Numér., 32, pp. 671-680; Krishnan, A., Kulkarni, S.H., Pseudospectrum of an element of a Banach algebra (2017) Oper. Matrices, 11, pp. 263-287; Reddy, S.C., Pseudospectra of Wiener–Hopf integral operators and constant-coefficient differential operators (1993) J. Integral Equ. Appl., 5, pp. 369-403; Seidel, M., On (N, ϵ) -pseudospectra of operators on Banach spaces (2012) J. Funct. Anal., 262, pp. 4916-4927; Shargorodsky, E., On the level sets of the resolvent norm of a linear operator (2008) Bull. Lond. Math. Soc., 40, pp. 493-504; Trefethen, L.N., Pseudospectra of linear operators (1997) SIAM Rev., 39, pp. 383-406; Trefethen, L.N., Embree, M., Pseudospectra gateway, , http://www.comlab.ox.ac.uk/pseudospectra, Web site; Trefethen, L.N., Embree, M., (2005) Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, , Princeton University Press, Princeton}, correspondence_address1={Dhara, K.; Department of Mathematics, India; email: kousik.dhara1@gmail.com}, publisher={Springer Basel}, issn={0378620X}, language={English}, abbrev_source_title={Integr. Equ. Oper. Theory}, document_type={Article}, source={Scopus},