I'm working on a question of Lang's book Complex Analysis. Let $[a,b]$ an $[c,d]$ be two intervals, not reduced to a point. Show that there is a function $g(t)=rt+s$ such that $g$ is strictly increasing, $g(a)=c$ and $g(b)=d$. Thus a curve can be parametrized by any given interval.
I was thinking that maybe $r>0$ for $g$ to be strictly increasing, but then $t>0$ has to be true as well. So now I'm stuck. Is there anyone who can help me? Thanks!
On
A simple way to see how this works is to think that the function $g(t)=rt+s$ is the equation of the segment that connect the points that have as coordinates the extremes of the two intervals (see the figure).
This equation, going from $A$ to $B$ is:
$$ y=g(t)=\frac{d-c}{b-a}(t-a)+c \qquad t\in[a,b] $$
Let's be systematic. An affine function $g(t)=rt+s$ is strictly increasing iff $r\gt 0$ . (Indeed $g(t_1)-f(t_2)=r(t_1-t_2)$). Now we need $g(a)=c$ and $g(b)=d$. This gives the system
$$\left\{ \begin{array}{c} ra +s=c\\ rb +s=d \end{array} \right.$$
Assuming $a\lt b$ and $c\lt d$, we have
$$r={d-c\over b-a}\gt 0$$
And $s=c-ra$. And we're done