EDB — 1GD

[1GD] Let \(f:A⊆ℝ^ n→ℝ\) be continuous, with \(A\) open, and let \(\overline x=(\overline x',\overline x_ n)∈ A\) be such that \(∂_{x_ n} f\) exists in a neighborhood of \(\overline x\), is continuous in \(\overline x\) and \(∂_{x_ n}f(\overline x) ≠0\). Define \(\overline a=f(\overline x)\).

There is then a ”cylindrical” neighborhood \(U\) of \(\overline x\)

where

is the open ball in \(ℝ^{n-1}\) centered in \(\overline x'\) of radius \(𝛼{\gt}0\), and

with \(𝛽{\gt}0\). Inside this neighborhood \(U∩ f^{-1}(\{ \overline a\} )\) coincides with the graph \(x_ n=g(x')\), with \(g:U'\to J\) continuous.

This means that, for every \(x=(x',x_ n)∈ U\), \(f(x)=\overline a\) if and only if \(x_ n=g(x')\).

Moreover, if \(f\) is of class \(C^ k\) on \(A\) for some \(k∈ℕ^*\), then \(g\) is of class \(C^ k\) on \(U'\) and