The derivatives of a function at a boundary point
I have a function
Typically the derivative is only defined on interior points of an interval where the function is defined. How can I rigorously argue that I can actually use
- If I consider
as the entire space, then the best linear approximation to at just happens to be the usual derivative. - If I can argue for analytic continuation of
to a slightly larger, open interval, then all's fine. - Or maybe calculate
and take .
Are these feasible tactics to use here? When are they a good idea? Any others that could help here?
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The endpoint fails to be an "interior" point only when the space
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Your question is better phrased and answered in Whitney's paper: Analytic extensions of differentiable functions defined in closed sets.
Here is an excerpt from the introduction section of the paper, which can be seen from the given AMS link:
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