It is well known that given $X,Y$ arbitrarily topological spaces, $I$ the unit interval, and continuous functions $f, g : X \rightarrow Y,$ a homotopy between the functions is a continuous function $H : X \times I \rightarrow Y$, such that $H(x,0) = f(x)$ and $H(x,1) = g(x)$.

It is stated in a lot of textbooks that one can think of such an homotopy as a one-parameter family of functions $\{H_{t}\}$, given by $ t \mapsto H_{t}$, where $H_{t}(x) := H(x,t)$.

Given that $I$ is a compact space it is clear that the continuity of the homotopy implies the continuity of the later map. But in general the opposite implication might not follow through, which takes me to my question:

Does anyone have a counterexample of a continuous function $\omega: I \rightarrow M(X,Y) $, where $M(X, Y)$ denotes the space of continuous functions with the compact-open topology, such that the induced map $H: X \times I \rightarrow Y$ given by $\omega (t)(x) := H(x,t)$, is not continuous?