This is from Apostol's Calculus Vol. I, Section 14.13 #21:
Let $C$ be a curve described by two equivalent functions $X$ and $Y$, where $Y(t)=X[u(t)]$ for $c\le t\le d$. If the function $u$ which defines a change of parameter has a continuous derivative in $[c,d]$ prove that
$\int_{u(c)}^{u(d)} \! ||X'(u)||\,\mathrm du=\int_c^d \! ||Y'(t)||\, \mathrm d t$ and deduce that the arc length of $C$ is invariant under such a change of parameter.
I believe that the condition placed on the derivative of $u$ should not have been continuity but rather non-negativity. First a counter-example: $Y(t)=t\boldsymbol i\,,\quad X(t)=-t\boldsymbol i \, , \quad u(t)=-t\,.$ Then $Y(t)=X[u(t)]$ over, say, $0\le t\le 1$ and $u'(t)=-1$ is certainly continuous. Now $||X'(t)||=1$ and $||Y'(t)||=1$ but $\int_{u(0)}^{u(1)} \! ||X'(u)||\,\mathrm du=\int_0^{-1}\!\mathrm d u=-1$ and $\int_0^1 \! ||Y'(t)||\,\mathrm dt=\int_0^1\!\mathrm d t=1\,.$
On the other hand, if we require $u'(t)\ge0$ (and I don't think we even need continuity, do we?) then we can write
$\begin{align}Y'(t)&=X'[u(t)]u'(t)\\ ||Y'(t)||&=||X'[u(t)]u'(t)||\\ &=||X'[u(t)]||\cdot |u'(t)|\\ &=||X'[u(t)]||u'(t)\\ \implies\int_c^d\!||Y'(t)||\,\mathrm d t&=\int_c^d \!||X'[u(t)]||u'(t)\,\mathrm d t\\ &=\int_{u(c)}^{u(d)}\!||X'(u)||\,\mathrm d u \end{align}$
Is this correct? Should the condition on $u'$ be non-negativity, rather than continuity, or do I need non-negativity in addition to continuity? I don't see anything in my proof at the end that requires continuity, but maybe I'm glossing over it.