Let $f:[a,b] \rightarrow \mathbb{R^n}$ and $g:[c,d] \rightarrow \mathbb{R}^n$ be equivalent paths in $\mathbb{R}^n$. Here, we are considering $f$ and $g$ to be equivalent if there exists a continuous and strictly monotonic bijection $\phi:[a,b] \rightarrow [c,d]$ such that $ f(t) = (g \circ \phi)(t) \; \; \forall \; \; t \in [a,b] $
Apostol in Mathematical Analysis (2nd Ed) claims on p 136 that equivalent paths have the same graph, but I can't possibly see how this can be so. For example,
$ \mathrm{graph}(g) = \{(u, g(u)) | u \in [c, d] \} = \{ (\phi(t), g(\phi(t)) | t \in \phi^{-1}[c,d] \} = \{(\phi(t), f(t)) | t \in [a,b] \} $ which is almost the graph of $f$ but because of the first coordinate is not the graph of $f$. I can only believe that I'm misunderstaning what he means by "graph". If "graph" of $f$ is defined as the image of $f$ then, yes, the statement would be true, but that's not how the graph of a map is defined.
Any ideas about what's going on here?