Let $G$ be a locally compact group and let $f_1,f_2 \in C_c(G)$ be nonnegative functions. I am working through a proof of a theorem and it in the author claims that we can choose $f'$ such that $f' \equiv 1$ on the support of $f_1 + f_2$. I am trying to justify why can choose such a function. I think it's pretty clear in the case when $G = \Bbb{R}$, but the general case seems hard (geometrically, it should be true). My thought was to take an open set containing the support of $f_1+f_2$, or a sequence of open sets containing, and letting the function "taper off" on this open set (or sequence of open sets). But it isn't clear to me how I should do this.
EDIT:
I think this is the relevant version of of Urysohn's lemma.
Let $X$ be a locally compact Hausdorff space, $K$ compact, and $A$ closed s.t. $K \cap A = \emptyset$. There exists a continuous function of compact support $f : X \to [0,1]$ with $f = 1$ on $K$ and $f = 0$ on $A$.
And I think the proof should go like this. Since $G$ is not compact (otherwise take $f' = 1$ on $G$), there is a point $g_0 \in G \setminus supp(f_1 + f_2)$. Then $A := \{g_0\}$ is a closed set disjoint from $supp(f_1 + f_2)$, so Urysohn's lemma states that there is a continuous function $f : G \to [0,1]$ with compact support such that $f = 1$ on $supp(f_1 + f_2)$ and $f(g_0) = 0$.