Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain. In this case, the theorem gives a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

For functions of a single variable, the theorem states that if ${\displaystyle f}$ is a continuously differentiable function with nonzero derivative at the point ${\displaystyle a}$, then ${\displaystyle f}$ is invertible in a neighborhood of ${\displaystyle a}$, the inverse is continuously differentiable, and

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