Derivative of floor function using epsilon/delta
Let $\delta$ be the smallest of the distances from $a$ to $\lfloor a\rfloor$ and to $\lceil a\rceil$; for instance, if $a=\frac53$, then $$\delta=\min\left\{\left|\frac53-1\right|,\left|\frac53-2\right|\right\}=\min\left\{\frac23,\frac13\right\}=\frac13.$$Then$$|x-a|<\delta\implies x\in\left(\lfloor a\rfloor,\lceil a\rceil\right)\implies\lfloor x\rfloor=\lfloor a\rfloor,$$and therefore$$\frac{\lfloor x\rfloor-\lfloor a\rfloor}{x-a}=0<\varepsilon.$$