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I'm reading Taylor theorem in textbook Analysis I by Amann.

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and its proof:

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Here is the Theorem 2.18 used in the proof:

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I read the proof carefully, but could not find where the authors apply the assumption that $D$ is convex. So I think that this assumption is unnecessary.

Could you please confirm if my understanding is correct? Thank you so much!

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    $\begingroup$ Look closely at the definition of the bound on the remainder function. It requires the line segment from $a$ to $x$ to be in $D.$ $\endgroup$ Sep 17, 2019 at 22:57

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It definitely comes into play in the definition of $h$, as you assure that $a + t(x-a) $ $\in D$

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