The question is in the title.I have tried many many things,that did not get me anywhere really,so it would be pointless to write them here.
Note: [x] denotes floor of x
Basically I need to prove that if a is not an integer then $$\lim_{x \to a} [x] = [a]$$
Any takers?
Hint: If $a\in\mathbf{R}$ is not an integer, there is small enough $\epsilon$ so that there is no integer in $(a-\epsilon,a+\epsilon)$.
Solution (sketch): Suppose that $[a]=n\in\mathbf{Z}$. The above statement tells us when we are close enough to $a$, say within $\epsilon$ of $a$ for some $\epsilon>0$, that $(a-\epsilon,a+\epsilon)\cap \mathbf{Z}=\emptyset$. So, for all $x$ close enough to $a$, that is, $x\in (a-\epsilon,a+\epsilon)$, $[x]=n$ and therefore, $|[x]-[a]|=|n-n|=0$.
I'll leave it to you to formalize to your desire, but that's about it. Hope it helps.