Extending the definition of curve length

Question

I know for continuously differentiable curves on closed interval $[a,b]$, the curve length is given by $\Lambda (\gamma)=\int_a^b |\gamma^{'}(t)|dt$.
But what about curves such that $\gamma^{'}(t)$ is only continuous on $[a,b)$ but not necessarily at $b$. Can we show that in this case $\Lambda (\gamma)=\lim_{x\rightarrow b}\int_a^x |\gamma^{'}(t)|dt$ ?

Answer 1

Some thoughts that are too long for a comment:

If you consider a function continuous on $[a,b]$ we must have some information about what the function does outside of $[a,b]$ as well in order to define the derivative at the endpoints. But, if we have that information and the derivative is not continuous only at $b$, then it must be only jump discontinuous at $b$.

Examples that come to mind here are $\gamma(t)=|t|$ on $[-1,0]$ but, if we restrict to this interval only and "forget" the information of the function beyond this interval, defining the derivative at the endpoints does not make much sense beyond some form of extension.

To "extend the derivative" we can define $\gamma^{\ast}(b):=\lim\limits_{t\rightarrow b}\gamma'(t)$ and $\gamma^*(t)=\gamma'(t)$ elsewhere.

Furthermore, we have $$\lim\limits_{x\rightarrow b}\int_a^x|\gamma'(t)|dt=\int_a^b|\gamma^*(t)|dt$$ since these two functions differ by only a point.

A similar claim will be true about functions on the closed interval $[a,b]$ with continuous derivative on the interval $(a,b)$.

Perhaps there a functions I am forgetting but, relaxing the conditions on your function too much could yield some functions which we can only approximate in arclength (or lead to differentiable nowhere functions with infinite perimeter). Maybe an analyst can give a more appropriate answer!