= Solution
For $n=1$, <differentiability> at $x_0$ means existence of the finite two-sided <limit of a function> $\lim_{h\to0}(f(x_0+h)-f(x_0))/h$. More generally, <higher-order differentiability at a point> is defined recursively: for $n\geq2$, the <derivatives> $f',\ldots,f^{(n-1)}$ must exist throughout some open interval containing $x_0$, and $f^{(n-1)}$ must be differentiable at $x_0$. With $f^{(0)}=f$, the final condition is
$$
\boxed{f^{(n)}(x_0)=\lim_{h\to0}
\frac{f^{(n-1)}(x_0+h)-f^{(n-1)}(x_0)}h\in\mathbb R.}
$$
The neighbourhood requirement makes the numerator meaningful for every sufficiently small nonzero $h$. It does not require $f^{(n)}$ to exist at other points of that neighbourhood, or to be continuous at $x_0$.
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