Geodesic
On Smooth Functions That Are Even on the Boundary of a Ball
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 156-161

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We show how to construct a smooth function without critical points on the ball $B^n$, $n>1$, that is even on its boundary $S^{n-1}$. In particular, it follows that the corresponding generalization of Rolle's theorem to dimensions $n>1$ does not hold. 
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     author = {S. E. Zhukovskiy and K. V. Storozhuk},
     title = {On {Smooth} {Functions} {That} {Are} {Even} on the {Boundary} of a {Ball}},
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S. E. Zhukovskiy; K. V. Storozhuk. On Smooth Functions That Are Even on the Boundary of a Ball. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 156-161. http://geodesic.mathdoc.fr/item/TM_2023_321_a9/