Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$.
The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map $c_x$ (where $\bar \alpha =\alpha(1-t)$ is:
$F(s,t) = \alpha(2s)$, if $0\le s \le t/2$
$F(s,t) = \alpha(2s)$, if $t/2 \le s \le 1- (t/2)$
$F(s,t) = \alpha(2s)$, if $1-(t/2)\le s \le 1$
By the gluing lemma this function is continuous.
I'm wondering, why we can't use this function instead:
$H(s,t) = c$ , if $0\le t \lt 1$
$H(s,t) = \alpha \bar \alpha(s)$, if $t=1$
Yes, I know H is not continuous, but how can I prove this formally?