I am trying to find the continuity of $[x]-[-x]$ where $x\in\Bbb{R}$ and $[\;]$ denotes the greatest integer function. Here is what I have done;
Let $f(x)=[x]-[-x].$ It is obvious that if $b$ is an integer, then
$$\lim_{x\to b-} f(x)=\lim_{x\to b-}[x] -\lim_{x\to b-}[-x] =(b-1)-[-(b-1)]=2b-2.$$ Also, $$\lim_{x\to b+} f(x)=\lim_{x\to b+}[x] -\lim_{x\to b+}[-x] =b-(-b)=2b,$$ and $$f(b)=[b] -[-b] =b-(-b)=2b.$$
This shows that $f(x)$ is not continuous for all integer values of $x$. Is it also discontinuous for all $x\in\Bbb{R}?$
I would like to know if I'm right. If yes, can anyone also help extend it to $\Bbb{R}?$ I would be very glad for that help.
Just in case my comment wasn't clear, your calculations are incorrect, though your conclusion is right. The first line should read $$ \lim_{x\to b-} f(x)=\lim_{x\to b-}[x] -\lim_{-x\to-b+}[-x] =(b-1)-(-b)=2b-1$$
Now correct the second line similarly.
$$\lim_{x\to b+} f(x)=\lim_{x\to b+}[x] -\lim_{-x\to -b-}[-x] =b-(-b-1)=2b+1$$
Compare this to the graph (of the right function this time!) on Wolfram Alpha.