Suppose that $f$ is the continuous function from $X^2$ to $I=[0,1]$ and $K$ is the compact subset of $X$. Define the function $F$ from $X$ to $I^K$ by $F(x)(b)=f(x,b)$. The topology on the space $I^K$ is compact-open topology. Is $F$ continuous?
Theorem 3.4.1 of Engelking’s General Topology says that for every pair of topological spaces $X$ and $Y$, the compact-open topology on $Y^X$ is proper. Just before Proposition 2.6.11 he defines the notion of a proper topology on $Y^X$ as follows:
Take his $X,Y,Z$ to be your $K,I,X$, respectively, and you have the desired result: your $F$ is continuous.