Journal of Convex Analysis 32 (2025), No. 1, 263--290
Copyright Heldermann Verlag 2025
Derivative of Inverse Mapping and Chain Rule Formulas for Gateaux Derivatives, Theorems and Counter-Examples
Marián Fabian
Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic
fabian@math.cas.cz
Jan Kolar
Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic
kolar@math.cas.cz
[Abstract-pdf]
We investigate what happens if Fr\'echet differentiability is replaced by Gateaux differentiability in several statements from differential calculus. Our main counter-example shows how the formula for derivative of inverse mapping fails when Gateaux differentiability is assumed both for the mapping and its inverse: We construct an involution $f: \mathbb{R}^2 \longrightarrow\mathbb{R}^2$ with $f(0,0)=(0,0)$, $C^\infty$-smooth off the origin and Gateaux differentiable at $(0,0)$, whose derivative $f'(0,0)$ is singular, and hence $f'(0,0)\,{\circ}\,f'(0,0)$ is not the identity mapping. This mapping simultaneously provides a strong counter-example for the Chain Rule formula with Gateaux derivatives. We also show that the chain rule formula holds true for the Gateaux derivatives in an important special case of mappings between Banach spaces.
Keywords: Chain rule formula, inverse mapping theorem, derivative, Frechet differentiability, Gateaux differentiability, involution.
MSC: 26B10, 46G05, 49J50, 58C20.
[ Fulltext-pdf (216 KB)] for subscribers only.