We say that paths $\alpha, \beta: I \to X$ with common initial point $\alpha(0)=\beta(0)$ and common terminal point $\alpha(1)=\beta(1)$ are homotopic provided that there is a continuous function $H:I \times I \to X$ such that $H(t,0)=\alpha(t)$ and $H(t,1)=\beta(t)$ for $t \in I$ and $H(0,s)=\alpha(0)=\beta(0)$ and $H(1,s)=\alpha(1)=\beta(1)$ for $s \in I$.
Since a path is a continuous function, I would expect that each map between two homotopic paths, $\alpha$ and $\beta$, is a path, and therefore continuous.
My questions are: is the above true, is it guaranteed by continuity of $H$, and if so, how do we prove it?
On a related note, intuitively we pass from $\alpha$ to $\beta$ as t varies over $[0,1]$. In general, can we explicitly realize a map 'in between' $\alpha$ and $\beta$ by fixing a $t$?