Proving an IVT has a local unique solution

Question

Given a continuous function $f\colon (a,b)\rightarrow{\mathbb{R}}$ and $x_0\in (a,b)$. Consider de initial value problem $$ \begin{cases} x'=f(x)\\ x(t_0)=x_0, \end{cases}$$ with $f(x_0)\neq 0$. How do I prove that this IVT has a unique local solution? I would prove that $f$ is locally Lipschitz, but I am not given the function, so I cannot see that.