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单词 inverse function theorem
释义

inverse function theorem

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.

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更新时间:2025/4/29 16:22:21