The derivatives of a function at a boundary point
I have a function $f \colon [0, L[ \, \to \mathbb{R}$ and I want to use the derivatives of arbitrary high orders of this function at zero. The function is defined on the half-open interval $[0, L[ \, \ni 0$, for an $L >0$.
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 $f^{(n)}(0)$ for all, or all high, values of $n$?
- If I consider $[0, L[$ as the entire space, then the best linear approximation to $f$ at $0$ just happens to be the usual derivative.
- If I can argue for analytic continuation of $f$ to a slightly larger, open interval, then all's fine.
- Or maybe calculate $f'(\varepsilon)$ and take $\varepsilon \to 0$.
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 $\left[0,L\right[$ is embedded in a larger space. If a function is defined on a certain domain, one may take the value of the derivative of the function $f$ at a point $a$ in the domain to be $$ f'(a) = \lim_{x~\to~a \atop x~\in~\text{domain}} \frac{f(x)-f(a)}{x-a}. $$
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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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