Curves in $\mathbb{R}^n$. Find a continuous bijective function so that an equation holds. Problem:
Let $p,q \in \mathbb{R}^n$ and let $\gamma: [a,b]\to\mathbb{R}^n$ ($a,b\in \mathbb{R}$ and $a<b$) a rectifiable curve with $\gamma(a) = p$ and $\gamma(b) = q$, which on no subinterval of $[a,b]$, that has positive length, is it constant.
Show that:
Let $L(\gamma) = |{p-q}|$ then there exists a continuous, bijective function $\delta:[a,b] \to [0,1]$ so that $$\gamma(t) = p + \delta(t)(q-p)$$ holds for all $t \in [a,b]$.
How do you prove this ? Proof ideas are welcome.

Take $$\delta(t)=\dfrac{t-a}{b-a},$$ for which one gets
$$\delta^{-1}(s)=(b-a)s+a,$$
that can be deduced by solving $t$ from $s=\dfrac{t-a}{b-a}$.
