Univalent function: Difference between revisions

Revision as of 22:23, 26 November 2008

Revision as of 02:18, 8 June 2010

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For [[real number|real]] [[analytic function]]s, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

:<math>f: (-1, 1) \to (-1, 1) \, </math>

given by <math>f''&fnof;''(''x'')&nbsp;=&nbsp;''x^''<sup>3</mathsup>. This function is clearly one-to-one, however, its derivative is 0 at <math>''x''&nbsp;=&nbsp;0</math>, and its inverse is not analytic, or even differentiable, on the whole interval <math>&nbsp;(-&minus;1, &nbsp;1).</math>

== References==