Suppose $(X,d)$ is a metric space and let $\mathcal{L}$ be the space of bounded Lipschitz functions on $X$. Let $D$ be a countable dense subset of $X$ and consider the set of functions $$h_{q_1,q_2,k,y}(x)=\min\{(q_1+q_2d(x,y)),k\}, \ q_1, q_2, k \in \mathbb{Q}, \ q_2,k\in(0,1), \ y \in D$$ and let $\mathcal{D}$ the set generated by these functions by taking $\inf$ over finitely many of them. Let $\mathcal{C}=\{\lambda f\ | \ \lambda\in\mathbb{Q}, f\in \mathcal{D}\}$. Is it true that $\mathcal{C}$ is dense (with the uniform metric) in $\mathcal{L}$? If so, how can I prove it? Hints or references are also greatly appreciated.
UPDATE copper.hat pointed out this is false when $X=\mathbb{R}$. Is it true if $(X,d)$ is compact?
(Reference: page 107 of this book: http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8721-1 )
Let $X=\mathbb{R}$ with the usual metric.
First take a representative $h(x) = \min(a+b|x-y|, c)$, where $b \ge 0$. If $b=0$, then $h$ is a constant, if $b \neq 0$ then for $|x|$ sufficiently large we see that $h$ is constant.
We then see that if $d \in {\cal D}$ then for $|x|$ sufficiently large, $d$ is a eventually a constant.
Hence, if $h \in {\cal C}$, then it is eventually a constant.
Choose $f(x) = \min_{k \in \mathbb{Z}} |x-2k|$. Note that $f(2n) = 0, f(2n+1) = 1$ for all integers $n$.
Suppose $h \in {\cal C}$ such that $\|f-h\|_\infty < {1 \over 4}$, in particular $h$ is eventually a constant, which gives a contradiction.