I'm reading Rourke and Sanderson's book "Introduction to Piecewise-Linear Topology", and I tried to show that the composition of two p.l. maps is p.l. (this is an exercise 1.6(2)) , but I couldn't. I think if any p.l. map is always continuous, then I can show that. So what I want to ask is p.l. map is always continuous? And if it doesn't hold, how can I show that the composition of two p.l. maps is p.l.?
(These are the definitions in the book:
For $a\in \mathbb{R}^m, L\subset\mathbb{R}^m$, $aL:=\{(1-t)a+tb; b\in L, t\in[0,1]\}$ is a cone if each point $x\in aL$ which is not $a$ is uniquely expressed by the form $(1-t)a+tb$ ($b\in L, t \in [0,1]$).
$P\subset\mathbb{R}^m$ is a polyhedron if each point $a\in P$ have its cone neighborhood $N=aL\subset P$ with $L$ compact.
$f:P\to Q$ ($P\subset\mathbb{R}^m,Q\subset\mathbb{R}^n$ are polyhedra) is p.l. if each point $a\in P$ have its cone neighborhood $N=aL$ ($L$:compact) such that $f((1-t)a+tb)=(1-t)f(a)+tf(b)$ for any $b\in L,t\in[0,1]$.)
Yes, PL maps are always assumed to be continuous. If you read page 1 of their book, they say "a map is a continuous function." It is very common among topologists to use the word "map" only for continuous maps.