A theorem which guarantees the existence of a local inverse of a real function, which generalizes to multivariable functions. If the real continuously differentiable function f has non-zero derivative at a, then f has a continuously differentiable inverse in a neighbourhood of a which satisfies:
where f(a) = b. Similarly, the multivariate version states that if F:ℝn→ℝn is continuously differentiable and has invertible differential dFp at a point p, then F has a continuously differentiable inverse in a neighbourhood of p satisfying
where q = F(p).
The following result relating to continuous real functions might also be referred to as the inverse function theorem. If f:[a,b]→[c,d] is continuous, strictly increasing, and satisfies f(a) = c, f(b) = d, then f has a strictly increasing continuous inverse.