# Non-singular boundary point

regular boundary point

An accessible boundary point (cf. Attainable boundary point) $\zeta$ of the domain of definition $D$ of a single-valued analytic function $f ( z)$ of a complex variable $z$ such that $f ( z )$ has an analytic continuation to $\zeta$ along any path inside $D$ to $\zeta$. In other words, a non-singular boundary point is accessible, but not singular. See also Singular point of an analytic function.

Note that the same point in the boundary of $D$ may give rise to several different accessible boundary points, some of which may be singular, others regular. E.g., consider the domain $D = \mathbf C \setminus ( - \infty , 0 ]$, and the function $f ( z) = ( h ( z) - \pi i ) ^ {-} 1$, where $h$ is the principal value of $\mathop{\rm log} z$. Then "above" $- 1$ there are two accessible boundary points: one singular, corresponding to approach along $z = - 1 + i t$, $0 \leq t \leq 1$; one regular, corresponding to approach along $z = - 1 - i t$, $0 \leq t \leq 1$.