Remarks on the inverse and implicit function theorems for differentiable functions (Q2024024)

The authors provide detailed proofs of the inverse and implicit function theorems in Euclidean spaces for differentiable functions which are not necessarily continuously differentiable. Also, under a kind of boundedness condition of the derivative of a differentiable function \(F:W\subset \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n\), it is proved that there exists a unique solution for the initial value problem: \( F(t,x,x') =z_0\), \(x(t_0)=x_0\), \(x'(t_0)=y_0\).