# functional analysis – How to prove that support of \$u_{r}\$ is compact?

Let’s $${u_r}:{mathbb{R}^k}to{mathbb{R}}$$, $$varphi in C_{c}(mathbb{R}^n)$$, $$Jf$$ ( jacobian of $$f$$, with $$f: mathbb{R}^{k}to mathbb{R}^{n}$$ Lipschitz ), and $$chi_E$$ (characteristic function on $$E subset mathbb{R}^{k}$$ where $$E$$ borel bounded subset),
$$begin{equation*} u_r(w) := chi_E(z+rw)varphiBigl(frac{f(z+ rw)-f(z)}{r}Bigr) J f(z+rw). end{equation*}$$

Statement: there are $$r_0> 0$$ and $$R> 0$$ such that $$supp(u_{r}) subset mathbb{B}(0,R)$$ for $$r in (0, r_{0})$$.

My attempt:
In fact, since $$f$$ is derivable into $$z$$ and $$Jf(z)> 0$$, there are $$s_{0}, lambda > 0$$ such that
$$begin{equation} |{f(z’)-f(z)}| geq lambda ||{z’-z}|| end{equation}$$
for every $$z ‘in mathbb{B}(z,s_{0})$$. On the other hand, if $$rho> 0$$ is such that $$supp (varphi) subset mathbb{B}(0,rho)$$, then
$$begin{equation*} |{f(z+rw) -f(z)}|leq rrho end{equation*}$$
for all $$win supp(u_{r})$$. Hence, if $$w in supp(u_{r})$$ with $$r < s_{0} / rho$$, you have $$z + rw in mathbb{B}(z,s_{0})$$, so $$r rho geq |{f (z + rw) -f (z)}| geq lambda ||{z + rw-z}|| = lambda r ||{w}||$$, then $$||{w}|| leq rho / lambda$$, which proves statement with $$r_{0}: = s_{0} / rho$$ and $$R: = rho / lambda$$.

I’m not sure what $$z + rw in mathbb{B}(z,s_{0})$$ , I really appreciate it if someone could give me an idea how to improve this argument.

Another idea that I had, was to prove that the $$supp(u_{0})$$ is compact, which I did, in order to arrive at that the $$supp(u_{r})$$ is compact, which would be another way to conclude that statement in another way. But unfortunately I could not find that relationship between the $$supp(u_{0})$$ and the $$supp(u_{r})$$I really appreciate the attention given.