  We consider an equation of multiple variables in which a partial derivative
does not vanish at a point. The implicit function theorem provides a local
existence and uniqueness of the function for the equation. In this paper, we
propose an algorithm to approximate the function by a polynomial without using
higher-order differentiability, which depends essentially on integrability.
Moreover, we extend the method to a system of equations if the Jacobian
determinant does not vanish. This is a robust method for implicit functions
that are not differentiable to higher-order. Additionally, we present two
numerical experiments to verify the theoretical results.
