Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous function $f:X\times Y\to\mathbb{R}$ such that for each $x\in X$, the function $f_x:=f(x,\cdot)$ is in $C_0(Y)$, the space of continuous functions on $Y$ vanishing at the infinity, and there exists a bounded open neighbourhood $U(x)$ of $x$ such that $f$ is (uniformly) bounded on $U(x)\times Y$. I'm trying to show that if we define $\varepsilon(t,x,y)$ by $$ \varepsilon(t,x',y)=\frac{1}{t}\int_0^t((Q_sf_{x'})(y)-f_{x'}(y))ds $$ then $\varepsilon$ converges to $0$ as $t$ goes to $0$ uniformly on $x'\in U(x)$ (we may let $y$ fixed). Since $(Q_t)_{t\geq 0}$ is a Feller semigroup, we have pointwise convergence if we fix $x'$ and $y$, but I don't see how to use the exta hypothesis on $f$ to show the uniform convergence. By the estimate $$ |\varepsilon(t,x',y)|\leq \frac{1}{t}\int_0^t|(Q_sf_{x'})(y)-f_{x'}(y)|ds\leq \frac{1}{t}\int_0^t\sup_{x'\in U(x)}|(Q_sf_{x'})(y)-f_{x'}(y)|ds $$ I tried to show that the function $g(s,y)=\sup_{x'\in U(x)}|(Q_sf_{x'})(y)-f_{x'}(y)|$ is continuous in $s$ at $0$, but I couldn't get anything from this.
I think it may be solved by using the uniform continuity of $f$ in $U(x)\times\{y\}$, but still I don't see how to proceed.
Thanks in advance for any help.