A contour or path is a continuous mapping $\gamma:[a,b]\rightarrow \mathbb{C}$ which is piecewise continuously differentiable, i.e., there exist $a=a_0<a_1<...<a_n=b$ such that $\gamma_{|[a_{j-1},a_j]}$ is continuously differentiable for each $j$
What is the meaning of "$\gamma_{|[a_{j-1},a_j]}$" in this text?
In general, if $f:X\to Y$ and $A\subseteq X,$ then $f_{|A}:A\to Y$ is the “restriction of $f$ to $A.$”
Technically, we first define $i_{A}:A\to X$ as the natural inclusion function, then define $$f_{|A}=f\circ i_A.$$
Functions have a fixed domain, so the function $f$ has only one domain.
Functions also have only one co-domain (slightly different from the “range” of a function.) Thus is sometimes a little odd.
For example: $f(x)=x^2$ can be defined as a function $\mathbb R\to\mathbb R,$ or we could define it as a function $\mathbb R\to\mathbb R^{\geq 0}.$
Mathematicians play a little more fast and loose with the co-domains than they do with the domain, even though they are technically, trickier. Given a $B\subseteq Y,$ there isn’t always a corresponding function $f^{|B}:X\to B.$ (And that notation $f^{|B}$ is not, as far as I know, used at all, I just made it up for the sake of example.)