“Introductory real analysis” by Kolmogorov and Fomin defines continuous curves by the equivalence of continuous functions by the following.
Let $R$ be a metric space and suppose $a’≤b’$ and $a’’≤b’’$. Two continuous functions $f:[a’,b’]→R$ and $g:[a'',b'']→R$ are said to be equivalent if there exist two continuous nondecreasing functions $φ(t),ψ(t)$, defined on the same interval $[a,b]$ such that $φ(a)=a’,φ(b)=b’,ψ(a)=a'',ψ(b)=b''$, and $f(φ(t))=g(ψ(t))$ for all $t∈[a,b]$. ”
However, I don’t know why this relation satisfies transitivity. When $f$ and $g$ are equivalent and $g$ and $h$ are equivalent, I don’t know how to define $φ$ and $ψ$ corresponding to the equivalence between $f$ and $h$. Please help me.