I came across a proof regarding the relationship between the fundamental group and the first singular homology group over $\mathbb{Z}$ and there is a step I wish to clarify a bit.
Let $X$ be a topological space, I have two relative homotopic paths $\gamma \thicksim \gamma'\,\, rel\,\{0,1\}$ via the homotopy $F$, and I'm gonna call $x_0=\gamma(0)=\gamma'(0)$ the starting point and $x_1=\gamma(1)=\gamma'(1)$ the end point. Now I define the following map: \begin{align} \varphi:\,\, &I\times I \to \Delta_2 \\ &(s,t)\mapsto (s-st,st) \end{align} where $\Delta_2$ is the standard 2-simplex. This map can be restricted to $(0,1]\times I$ giving us the homeomorphism: $\varphi_{|}:= \varphi_{|(0,1]\times I}$. Now the proof define the following new map $\sigma:\Delta_2\to X$ defined as follows:
\begin{equation} \sigma(s,t):= \begin{cases} F\circ(\varphi_{|})^{-1}(s,t) & (s,t)\neq (0,0) \\ x_0 & (s,t)=(0,0) \end{cases} \end{equation} Now, the proof takes for granted the continuity of such $\sigma$, however it doesn't seem so obvious to me so here I am, I tried to prove the contiuity but failed so far.
I know this might be a naive question, but any help is very much appreciated :)