= Higher-order differentiability at a point
{title2=$f^{(n)}(a)$}
For $n=1$, <differentiability> at $a$ means that the <difference quotient> has a finite two-sided <limit of a function>. For $n\geq2$, require <derivatives> through order $n-1$ throughout some open <neighbourhood> of $a$, and require the final <difference quotient>
$$
f^{(n)}(a)=\lim_{h\to0}\frac{f^{(n-1)}(a+h)-f^{(n-1)}(a)}h
$$
to exist as a finite real number. The last <derivative> need not exist elsewhere or be continuous at $a$. In particular, existence of $f'(a)$ by itself does not justify taking a second <derivative>: the function $f'$ must first be defined in a <neighbourhood> of $a$.
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