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Toby Bartels (talk | contribs) →Comparison with real functions: Enlarging the ___domain. |
Toby Bartels (talk | contribs) m →Comparison with real functions: Thanks, BracketBot! |
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:<math>f: (-1, 1) \to (-1, 1) \, </math>
given by ''ƒ''(''x'') = ''x''<sup>3</sup>. This function is clearly one-to-one, however, its derivative is 0 at ''x'' = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the ___domain to an open subset <i>G</i> of the complex plane, it must fail to be one-to-one; and this is the case, since (for example) <i>f</i>(εω) = <i>f</i>(ε) (where ω is a [[primitive root of unity|primitive cube root of unity]] and ε is a positive real number smaller than the radius of <i>G</i> as a neighbourhood of 0).
== References==
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