Let $(M,g)$ be a complete Riemannian manifold . Let $f\in C^\infty(\mathbb{R})$ such that $f\equiv1$ on $(-\infty,0]$ and $f\equiv0$ on $[1,\infty)$ . Let $x_0\in M$ be fixed and $d_g(x,x_0)$ be the geodesic distance between $x,x_0$ for any $x\in M$ . Define $f_i:M\to\mathbb{R}$ by $$f_i(x)=f(d_g(x,x_0)-i)$$ for all $i\in\mathbb{N}$ . I want to prove $f_i$ are Lipschitz continuous and have compact support .
Since $f$ has bounded derivative , it is Lipschitz and so are $f_i$ . But how to prove the compact support ? Is it dependent on the completeness of $M$ ? Any help is appreciated .