\begin{definition}[Definition:Piecewise Regular Curve Segment]
Let $M$ be a smooth manifold.
Let $I := \closedint a b$ be a closed real interval.
Let $\gamma : I \to M$ be a curve segment. 
Let $\tuple {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ be a finite subdivision of $I$.
For all $i \in \N : i < n$ let $\closedint {x_i} {x_{i + 1} }$ be a subinterval of subdivision of $I$.
Suppose for all $i \in \N : I < n$ the curve segment $\valueat \gamma {\closedint {x_i} {x_{i + 1}} }$ is regular.
Then $\gamma$ is said to be a '''piecewise regular curve segment'''.
\end{definition}