Suppose \(F:\mathbb R^2\to\mathbb R^2\) is continuously differentiable, satisfies \[ F(0,0)=(0,0), \] and has Jacobian matrix equal to the identity at \((0,0)\). Outline a direct proof that there exists \(\delta>0\) such that whenever \[ a^2+b^2<\delta, \] there is a point \((x,y)\in\mathbb R^2\) with \[ F(x,y)=(a,b). \]
Do not merely restate the Inverse Function Theorem: the requested argument is part of its proof. You may use without proof basic estimates expressing that the change in \(F\) is approximated by its differential.