we find this definition on Karatzas [Brownian Motion and Stochastic Calculus pg 282]
But the definition of Kolmogorov Feller process does not seem to require that $Af$ be continuous in $x$ as is required in the definitions of Feller processes. for instance:
to be found on the site almost sure (https://almostsure.wordpress.com/2010/07/14/feller-processes/)
Therefore my question is: Shouldn't a Kolmogorov-feller process be at least a Feller process? I mean shouldn't we require that $Af$ is continuous on $x \in E$.
Are there any other authors that use the same definition of Kolmogorov-Feller process?