The situation is the following: Let be $G$ a locally compact (Hausdorff) group such that $G = H \rtimes_{\alpha} N$ is the semi direct product of locally compact groups $N$ and $H$. Let $A$ be a C$^*$-algebra and $f \in C_c(N,A)$ (continuous function with compact suport). I need to prove that $f$ is uniformly continuous in te following sense: Given $\varepsilon > 0$ there is a neighborhood $V$ of $1_H$ such that for all $h \in V$ we have: $$| f(h^{-1}nh) - f(n)| < \varepsilon,$$ for all $n \in N$.
I have a result that if $f \in C_c(G,A)$ then $f$ is right/left uniformly continuous, i.e., there is a neighborhood $V$ of $1_G$ such that if $h \in V$ we have $| f(hg) - f(g)| < \varepsilon,$ and $| f(gh) - f(g)| < \varepsilon,$ for all $g \in N$.
My attempt was to use this result adding $f(nh) - f(nh)$ and using the triangular inequality, but i don't know how exactly to apply this result because $nh$ is in $G$ but not in $N$ and $f$ is defined only in $N$. I was thinking if i could see $f$ in $C_c(G,A)$ just puting $f(n,h) = f(n)$. This is a continous function but, i think that the support is not compact... is that right? I'm confused now.