Function property on balls and scaling

Assume $X$ is a compact metric space and $Y$ is a metric space. Assume $F:X\rightarrow Y$ is a map.

Fix $L\geq 1$. Suppose for each $x\in X$, there exists an $\epsilon_x>0$ such that for every $w_1,w_2 \in B_{\epsilon}(x)$, $d(F(w_1),F(w_2))\leq Ld(w_1,w_2)$.

Is true this property holds on all $X$?

$B_{\epsilon_x}(x)$ is open ball center x radius $\epsilon_x$

Since $X$ compact there exists $K\geq L$ such that for $w_1,w_2\in X$, $d(F(w_1),F(w_2))\leq Kd(w_1,w_2)$.

But can we take $K=L$? $($L independent of $x$)