Level set

Given u = f(x) for f: ℝⁿ → ℝ, then for each c ∈ ℝ the level set relative to c of f is

 { x: f(x) = c } ⊂ ℝn (in the domain of f).

In other point these are points with the same altitude.

Theorem (Implicit Function Theorem). Let

 f:  ℝ2 → ℝ, f(a) = c, a ∈  ℝ2, c ∈ ℝ.

Suppose

 f is differentiable on an open disk D around a, and that 
 f′ is continuous on D. 

Suppose further

 f′(a) ≠ 0.