Problem: Consider $X$, a metric space and $I=[0,1]$. Let $C(I,X)$ denote the set of continuous paths in $X$. Assume $A\subseteq C(I,Y)$ and that are given a continuous function $\delta:X\rightarrow (0,\infty)$ with:
For each $f\in A$, there exists a constant $r_{f}>0$ and an an open set $U_{f}$ containing $f$ in $C(I,Y)$ with the property that for all $g\in U_f$, $d(g(x),g(0))\leq \delta(g(0))$ for all $0\leq x\leq r_f$.
Is it true that this implies that for every $g\in A$, $d(g(t),g(0))\leq \delta(g(0))$ for any $t\in [0,1]$?
Attempt: As $A\subseteq \bigcup_{f\in A}U_f$, let $\psi_{f}:A\rightarrow [0,1]$ be partition of unity subbordinate to $(U_f)_{f\in A}$. I think maybe if I define $\Psi$ to be $\Psi= \sum_{f\in A}\psi_{f}r_f$ maybe will work.
From definition of partitions of unity, $\Psi:$
$(1).$$0\leq \Psi \leq 1$
$(2)$. $supp(\psi_{f})\subseteq U_{f}$.
$(3).$ For every $h \in A$, $\sum_{f\in A}\psi_{f}(h)=1$
But I don't see how to complete, maybe this $\Psi$ doesn't work? Because it is not necessarry true that $\psi_f(g)=0$ for $g\neq f$.