Differentiability of $x \mapsto \dfrac{x}{\| x \|}$

Asked Sep 20 '22 at 10:35

Active Sep 20 '22 at 10:35

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Let $(E, \| \, \|)$ be a normed vector space, and let $\Phi: E \, \backslash \{ 0_E \} \to E, \, x \mapsto \dfrac{x}{\| x \|}$.

Is $\Phi$ differentiable ? I did try to look at $\Phi(x + h) - \Phi(x)$, but I do not see what to do ...

Thank you for your help.

asked Sep 20 '22 at 10:35

JackEight

3

It depends on the norm. For example, if $| \cdot |$ is the $1$-norm on $E = \Bbb{R}^2$, then $\Phi$ is not differentiable at any point along the $x$-axis. – Theo Bendit Sep 20 '22 at 10:42
Great, thank you ! – JackEight Sep 20 '22 at 10:52

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