Existence and uniqueness of a local solution for x' = f(t,x) using inverse functions

A condition on the function $f$ is given such that the scalar ordinary differential equation $x' = f(t,x)$ with initial condition $x(t_0) = x_0$ has a unique solution in a neighborhood of $t_0$. An example illustrates that this result can be used when other theorems which put conditions on the difference $f(t,x)-f(t,y)$ do not apply