How can we show that $C_c^\infty(\mathbb R)$ strongly separates points?

Question

Let $C_b(\mathbb R)$ denote the set of bounded continuous function from $\mathbb R$ to $\mathbb R$. We say that $M\subseteq C_b(\mathbb R)$

1. separates points $:\Leftrightarrow$ $$\forall x,y\in\mathbb R:x\ne y\Rightarrow\exists f\in M:f(x)\ne f(y)\tag1$$
2. strongly separates points $:\Leftrightarrow$ $$\forall x\in\mathbb R,\delta>0:\exists k\in\mathbb N,\left\{f_1,\ldots,f_k\right\}\subseteq M:\inf_{y\::\:d(x,y)\:\ge\:\delta}\max_{1\le i\le k}|f_i(x)-f_i(y)|>0\tag2$$

How can we show that $C_c^\infty(\mathbb R)$ strongly separates points?

It's clear that $C_c^\infty(\mathbb R)$ separates points.

Answer 1

Given $x\in\mathbb R$ and $\delta>0$, take a single function $f\in C^\infty_c(\mathbb R)$ with $f(x)=1$ and $\mathrm{supp}(f)\subset B(x,\delta)$.

Then $f(x)-f(y)=1$ for $d(x,y)\geq\delta$.