In Baues' Algebraic Homotopy he states that $G: X \times I \rightarrow Y$ is continuous if and only if $\overline{G}: X \rightarrow C(I,Y)$ is continuous where $C(I,Y)$ is the set of all continuous functions from $I$ to $Y$ with the compact open topology on it and $\overline{G}(x) = G\circ i_x $ with $i_x: I \rightarrow X \times I$ and $i_x(t) = (x,t)$. He claims this is a well known result however I haven't been able to find anything thus far. Maybe it is commonly phrased in a different way. Anyway here is how I have tried to prove it thus far.
($\implies$) Suppose that $G:X \times I \rightarrow Y$ is continuous. Let $U_{K,O} = \{ \sigma: I \rightarrow Y \in C(I,Y)| \sigma(K) \subset O \}$ and let $A := \overline{G}^{-1}(U_{K,O}) = \{x \in X | G\circ i_x(K) \subset O\} = \{x \in X | G(\{x\} \times K) \subset O\} $. We know that $G^{-1}(O)$ is open in $X \times I$. Let $p:X \times I \rightarrow X$ be the projection onto the $X$ coordianate. Then $p(G^{-1}(O)) \subset X$ is open and contains $A$...
My guess is that $p(G^{-1}(O)) = A$ but I cant seem to find a way to prove the other inclusion.