According to my book, a curve (continuous mapping) is called regular if it's differentiable and the derivative is never zero.
And generally, the length of a curve $\gamma:[a,b]\to \mathbb R^2$ is defined by $$L=\int_a^b \|\gamma'(t)\| dt.$$
Thus, in order to define the length, $\|\gamma'(\cdot)\|$ has to be integrable on $[a,b]$.
If $\gamma$ is regular, isn't the length of $\gamma$ necessarily defined ?
I think that even if $\gamma$ is regular, $\|\gamma'(\cdot)\|$ is not necessarily integrable ...