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From my Calculus textbook:

enter image description here

Usually the text is careful to make sure the description above a formula establishes all the necessary preconditions to exist for the formula to be true. I noticed here though that $f^ {(n+1)} $ is referred to, without any mention of a requirement that it exist (f being order n differentiable by itself doesn't imply n+1 differentiability). Does the continuity of all the earlier derivatives imply it or is there an unstated assumption?

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The statement says

$f^{(n)}$ is differentiable

so that next derivative exists

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  • $\begingroup$ doh, didn't read closely enough $\endgroup$ Feb 19 '18 at 18:30
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    $\begingroup$ @JosephGarvin No, you didn't. But you did try to, which is more than many students do. So a good question even though it had an easy answer. $\endgroup$ Feb 19 '18 at 18:31

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