A note on directional Lipschitz continuity in the Euclidean plane. (English) Zbl 1484.26015
Summary: We prove a stronger version of a conjecture stated in a paper from 2017 by J. M. Ash and S. Catoiu [Real Anal. Exch. 42, No. 1, 185–192 (2017; Zbl 1384.26036)] concerning relations between various notions of the Lipschitz property and differentiability in the Euclidean plane. We also provide an improved version of the main result of that paper.
MSC:
| 26B05 | Continuity and differentiation questions |
| 26A16 | Lipschitz (Hölder) classes |
| 26B35 | Special properties of functions of several variables, Hölder conditions, etc. |
Keywords:
\( \sigma \)-porosity; Baire categories; directional Lipschitz continuity; directional smoothness; porosityCitations:
Zbl 1384.26036References:
| [1] | J. M. Ash and S. Catoiu, Directional Differentiability in the Euclidean Plane, Real Anal. Exchange, 42(1) (2017), 185-192. · Zbl 1384.26036 |
| [2] | L. Zajíček, Hadamard differentiability via Gâteaux differentiability, Proc. Amer. Math. Soc. 143 (2015), 279-288. · Zbl 1316.46039 |
| [3] | J. L. Massera and J. J. Schäffer, Linear Differential Equations and Functional Analysis, I, Ann. of Math., 67(3) (1958), 517-573 · Zbl 0178.17701 |
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