I'm working on problem 13 of chapter 0 from Hatcher's Algebraic Topology. Here is the statement:
Show that any two deformation retractions $r_t^0$ and $r_t^1$ of a space $X$ onto a subspace $A$ can be joined be a continuous family of deformation retractions $r_t^s$, $0 \leq s \leq 1$, of $X$ onto $A$, where continuity means that the map $X \times I \times I \to X$ sending $(x,s,r,t)$ to $r_t^s(x)$ is continuous.
I came up with the following candidate: $$ H(x,s,t) = r_t^s(x) = \begin{cases} r_t^1(x) & \text{if} \ t \in [0,s] \\ r_{\frac{t-s}{1-s}}^0 \circ r_s^1(t) &\text{if} \ t \in (s,1]. \end{cases} $$ I would like to show that $H$ it is continuous. I understand that it is continuous away from $s = 1$, but I'm not sure if there are some examples of spaces $X$ for which $H$ would not be continuous. I think it might work for spaces where the sequential criterion for continuity holds. Note that there is a perfect solution of this problem here, but I would like to know if my example works as well, since I find this solution more geometrically intuitive.