I have seen the following notation in a book for the continuous function $f$:
$f(.,.):\Bbb R^d \times \Bbb R^d \to \Bbb R$ is uniformly continuous in the first argument on compacts w.r.t the second argument.
What is the meaning of it ? Is it that for a fixed $y$ the function $f(.,y): \Bbb R^d \to \Bbb R$ is uniformly continuous on compacts i.e. a single $\delta$ works for "all" the compacts ?
So, for some other $y$, $\delta$ will be different.
The exact intercept from the paper: "$(y,z,x) \in S \times U \times \Bbb R^d \to p(dw|y,z,x) \in P(S)$ is continuous map. Moreover, the continuity in the $x$ variable is uniform on compacts w.r.t the other variables." $P(S)$ is the space of probability measures on a complete separable metric space $S$
Every continuous function of two variables $f:X\times Y\to Z$ from (say Hausdorff) spaces into a metric space $Z$ is uniformly continuous on compact sets in the second argument when the first argument is held fixed. This means that the function $f_x:Y\to Z$ for fixed $x$ is uniformly continuous on each compact subset of $Y$ (which is true for any continuous function).