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correcting the the miseprinte about the derivative.
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Am I missing something? $(\Gamma^{-1})'(y) = \frac{1}{\Gamma '(x)} = \frac{1}{\Gamma(x) \cdot \psi(x)}$ for $y = \Gamma(x)$. The existence of $I$ now immediately follows from the continuity of $\Gamma '$$(\Gamma^{-1})'$.

Am I missing something? $(\Gamma^{-1})'(y) = \frac{1}{\Gamma '(x)} = \frac{1}{\Gamma(x) \cdot \psi(x)}$ for $y = \Gamma(x)$. The existence of $I$ now immediately follows from the continuity of $\Gamma '$.

Am I missing something? $(\Gamma^{-1})'(y) = \frac{1}{\Gamma '(x)} = \frac{1}{\Gamma(x) \cdot \psi(x)}$ for $y = \Gamma(x)$. The existence of $I$ now immediately follows from the continuity of $(\Gamma^{-1})'$.

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Dieter Kadelka
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Am I missing something? $(\Gamma^{-1})'(y) = \frac{1}{\Gamma '(x)} = \frac{1}{\Gamma(x) \cdot \psi(x)}$ for $y = \Gamma(x)$. The existence of $I$ now immediately follows from the continuity of $\Gamma '$.