Inverse function theorem: Difference between revisions

Revision as of 12:58, 15 September 2023

Revision as of 13:25, 15 September 2023

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==Statements==

For a functions of a single [[Variable (mathematics)|variable]], the theorem states that if <math>f</math> is a [[continuously differentiable]] function with nonzero derivative at the point <math>a</math>; then <math>f</math> is injective (or bijective onto the image) in a neighborhood of <math>a</math>, the inverse is continuously differentiable near <math>b=f(a)</math>, and the derivative of the inverse function at <math>b</math> is the reciprocal of the derivative of <math>f</math> at <math>a</math>:

<math display=block>\bigl(f^{-1}\bigr)'(b) = \frac{1}{f'(a)} = \frac{1}{f'(f^{-1}(b))}.</math><!-- Not sure the meaning of the following alternative version; if the function is already injective, the theorem gives nothing: An alternate version, which assumes that <math>f</math> is [[Continuous function|continuous]] and [[Locally injective function|injective near {{Mvar|a}}]], and differentiable at {{Mvar|a}} with a non-zero derivative, will also result in <math>f</math> being invertible near {{Mvar|a}}, with an inverse that's similarly continuous and [[Injective function|injective]] , and where the above formula would apply as well. -->