We can find the statement of the theorem for continuously differentiable $f$ here https://en.wikipedia.org/wiki/Implicit_function_theorem.
Its also interestingly remarked that if $f$ is $k$ times continuously differentiable than we can choose $U$ so that $g$ is $k$ times continuously differentiable. How can we prove this using the theorem? I need this statement to prove some equivalent characterizations for sub-manifolds. Thank you!