 Local existence for a scalar autonomous ordinary differential equation with continuous vector field

For continuous $\varphi :\mathbb{R}\to \mathbb{R}$, the initial-value problemhas a local continuously differentiable solution. If $\varphi (0)=0$, use the constant solution; otherwise locally invert $F(x)={\int }_0^{x}du/\varphi (u)$.