How to prove this proposition on page 75 of Jeffrey Strom's book "Modern classical homotopy theory"?
Every map from a $1$-dimensional CW complex to a convex space is homotopic to a piecewise linear map.
Definitions:
Piecewise linear: If $X$ is a $1$-dimensional CW complex and $Y \subseteq V$ (where $V$ is an $\mathbb{R}$-vector space) then, we say that a function $f : X \to Y$ is piecewise linear if the composite $I \xrightarrow{\chi} X \xrightarrow{f} V$ of $f$ with each characteristic map $\chi$ is linear.
For points $x, y$ in an $\mathbb{R}$-vector space $V$ , we call any linear combination $(1-t)x +ty$ with $t \in I$ a convex combination of $x$ and $y$. The function $\alpha : t \to (1 - t)x + ty$ parametrizes the unique constant speed path (with domain $I = [0, 1]$) starting at $x$ and ending at $y$, called the linear path from $x$ to $y$. A subset $Y \subseteq V$ is called convex if every convex combination of points in $Y$ is also in $Y$ .
I know that
Let $A$ be a convex subset of $\mathbb{R}^n$, endowed with the subspace topology, and let $X$ be any topological space. Then any two continuous maps $f, g : X \to A$ are homotopic.
Proof: Let $H(x,t)=(1-t)f(x)+tg(x)$, then $H(x,t)$ is continuous since $f(x)$ and $g(x)$ are continuous. And clearly $H(x,t)$ is a homotopy between $f$ and $g$.
But the proposition does not require continuity. Could you please help me? Thanks!
Edit: changed the link to the book.
Assume that $f:X\rightarrow C$ is a map from a $1$-dimensional CW complex $X$ into a convex subspace $C\subseteq V$ of a real vector space $V$. Answering the question as written isn't really very interesting, since any two maps into a convex space are homotopic and it is not difficult to see that some PL map exists (by translation we can assume $0\in V$. Let $g:X\rightarrow C$ be constant at $0$). Rather we will show:
To this end define $g:X\rightarrow C$ by letting $g|_{X_0}=f|_{X_0}$ and requiring that whenever $\chi:I\rightarrow X$ is the characteristic function of a 1-cell of $X$, then $$g\circ\chi(t)=(1-t)\cdot g(\chi(0))+t\cdot g(\chi(1)),\qquad t\in I.$$ Then $g$ is well-defined and continuous, and is PL by definition. We get a homotopy relative to $X_0$ from $f$ to $g$ by letting $$F_s|_{X_0}=f|_{X_0},\qquad \forall s\in I$$ and requiring that $$F_s(\chi(t))=(1-s)\cdot g(\chi(t))+s\cdot f(\chi(t))$$ whenever a cell $\chi:I\rightarrow X$ is given. Then $F$ is a homotopy with the required properties. If you are worried about the continuity of $F$, give $X\times I$ the product CW structure. Since $I$ is locally compact, this is again a CW complex. Then clearly $F$ is continuous on each product cell of $X\times I$, and so continuous over all of $X\times I$.
p.s. I think rather than 'linear', a better term for the type of map $I\rightarrow C$ you are considering is 'affine'.
p.p.s. I think it's hilarious, but your link might not be entirely appropriate.