Smooth functions
- Tolaso J Kos
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Smooth functions
Let \( f:\mathbb{R}^n \rightarrow \mathbb{R} \) be a smooth function (meaning that it has continuous partial derivatives of any order).
Prove that there exist smooth functions \( g_1, \dots, g_n \) such that:
$$f\left ( x_1, \dots, x_n \right )-f\left ( 0, \dots, 0 \right )=\sum_{i=1}^{n}x_i g_i \left ( x_1, \dots, x_n \right )$$
Prove that there exist smooth functions \( g_1, \dots, g_n \) such that:
$$f\left ( x_1, \dots, x_n \right )-f\left ( 0, \dots, 0 \right )=\sum_{i=1}^{n}x_i g_i \left ( x_1, \dots, x_n \right )$$
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