In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if the best linear approximation to the function at a point is invertible, then with sufficient regularity assumptions, the function should also be invertible near that point. In its simplest form, the theorem states that if a real function f is differentiable in an open interval, with a continuous derivative, then in a neighborhood of any point where the derivative is not zero, f has an inverse function. The inverse function is also continuously differentiable, and the inverse function rule expresses its derivative as the multiplicative inverse of the derivative of f. The theorem applies verbatim to complex-valued functions of a complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between vector spaces of the same finite dimension, by replacing "derivative" with "Jacobian matrix" and "nonzero derivative" with "nonzero Jacobian determinant". If the function of the theorem belongs to a higher differentiability class, the same is true for the inverse function. There are also versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth. The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem.
Statements
One variable The inverse function theorem is not often stated separately for one variable, because a stronger result is true: if a real-valued function of a single real variable has a derivative which is nonzero on an interval, then the function has an inverse throughout the interval. More precisely, suppose that f {\displaystyle f} is a real-valued differentiable function on an open interval I {\displaystyle I} , and f ′ {\displaystyle f'} is non-zero throughout I {\displaystyle I} . Then the image of the interval I {\displaystyle I} is another interval J {\displaystyle J} , f : I → J {\displaystyle f:I\to J} is a bijection, and has a differentiable inverse function f − 1 : J → I {\displaystyle f^{-1}:J\to I} . This theorem is true because the non-vanishing of the derivative implies that it must be entirely of one sign (positive or negative) according to Darboux's theorem, and therefore the function must be strictly monotone, and thus one-to-one. The inverse function theorem is a weaker local statement. The statement of the inverse function theorem is, roughly speaking, that if a real-valued function f {\displaystyle f} of a single real variable has a continuous derivative on an open interval I {\displaystyle I} , then it is locally invertible near each point where its derivative is non-zero. More precisely, around any point x {\displaystyle x} in I {\displaystyle I} where f ′ ( x ) ≠ 0 {\displaystyle f'(x)\neq 0} , there is a smaller interval I ′ {\displaystyle I'} on which the function f {\displaystyle f} is one-to-one and the image of the interval I ′ {\displaystyle I'} under f {\displaystyle f} is also an open interval J ′ {\displaystyle J'} . Thus f {\displaystyle f} maps the interval I ′ {\displaystyle I'} bijectively onto the interval J ′ {\displaystyle J'} . The inverse function theorem in this form follows at once from the stronger global result, by restricting the interval I {\displaystyle I} to a smaller interval I ′ {\displaystyle I'} around x {\displaystyle x} on which the derivative is non-zero throughout. When f {\displaystyle f} is continuously differentiable, the inverse function f − 1 : J ′ → I ′ {\displaystyle f^{-1}:J'\to I'} is also continuously differentiable, and its derivative at any point y {\displaystyle y} in the interval J ′ {\displaystyle J'} is given by the inverse function rule:
d d y f − 1 ( y ) = 1 f ′ ( f − 1 ( y ) ) . {\displaystyle {\frac {d}{dy}}f^{-1}(y)={\frac {1}{f'(f^{-1}(y))}}.}
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