Suppose $\alpha : [0,l] \rightarrow S$ is a continuous map from closed interval $[0,l]$ to regular surface $S$. We say that $\alpha$ is piecewise regular parametrized curve if there exists a subdivision $0=t_0 < t_1 < ....< t_k < t_{k+1} = l$ of $[0,l]$ such that $\alpha$ is differentiable and regular in each $[t_i , t_{i+1}]$ for every $i = 0,1,...,k$.
I have some trouble in the last part of this definition. Piecewise regular curves fails to have a well-defined tangent line only at a finite number of points. Here these points are $t_0, t_1, .. , t_{k+1}$. So shouldn't we defining the regularity of $\alpha$ in open interval $(t_i, t_{i+1})$ rather than $[t_i, t_{i+1}]$