9. Functions of several variables

• $L(\mathbb{R}^{n}, \mathbb{R}^{n})$
  • linear mapping of $\mathbb{R}^{n}$ into $\mathbb{R}^{m}$,

Definiton 9.20

• $E \subseteq \mathbb{R}^{n}$,
• $f:E \rightarrow \mathbb{R}^{m}$,
  • differentiable

$f$ is said to be continuously differentiable in $E$ if $f^{\prime}$ is a contunous mapping of $E$ into $L(\mathbb{R}^{n}, \mathbb{R}^{n})$. That is

\[\forall \epsilon > 0, \ \exists \delta > 0, \ \text{ s.t. } \ y \in E, \ |x - y| < \delta, \Rightarrow \| f^{\prime}(y) - f^{\prime}(x) \| < \epsilon .\]

$f$ is said to be $\mathscr{C}^{\prime}$-mapping and $f \in \mathscr{C}^{\prime}(E)$.

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