Consider finite discrete sets $X$ and $A$ and a kernel $c:X\times A\to X$. As $X$ and $A$ are discrete sets endowed with discrete topology, we know that the function $c$ is continuous (in the strong sense).
Consider the following definition of weak continuity: a kernel $c$ is weakly continuous if the map $(x,a)\to\int_X f(z)c(dz|x,a)$ is continuous for all $f\in C_b(X)$.
We know that if $c$ is linear there exists an equivalence between weak and 'strong' continuity. However, $c$ might not be linear in this case.
As we are working on finite discrete sets, can we derive the following implication: $c$ continuous implies that $c$ weakly continuous ? If not, is there any condition for such results ?
Thank you for your help.