# Arc-Length Function

## Main Question or Discussion Point

I have the arc-length function defined for a smooth curve g: [a,b] -> R^n as starting at g(t_0) as:

s(t) = integral [t_0, t] ||dg/du|| du.

The text says this is differentiable, so ds/dt = ||dg/du||. But I don't see why. I know that g is smooth, but the norm causes problems and so to apply the fundamental theorem of calculus I would have to know that ||dg/du|| is continuous. If g is also regular then ||dg/du|| is smooth, so it would follow, but I don't see how this follows if g is simply smooth.

Any help?

But wait, the norm function is continuous so ||dg/du|| is continuous. Anyone confirm this?

arildno
Homework Helper
Gold Member
Dearly Missed
It is not that the norm function is continuous as such, but in conjunction with the fact that the derivative of a smooth function is continuous that the result follows..

Yes of course. My problem was that I know that dg/du is smooth, hence continuous, but once you apply the norm it may not be necessarily be true that it is continuous anymore. But since the norm is continuous it does follow now as the composition of two continuous functions.