For what values of a does lim_x-->a f(x) exist

[karush]

View attachment 1664

Basically I am trying to understand this question,

the graph is $$\lim_{x to a} {([[x]]+[[-x]])}$$

the last $$2$$ lines are the answers from W|A.

First, is looks like an greatest integer function, or notated as the floor function
next I presume $$\displaystyle x\rightarrow\text{a}$$ is where on the $$x$$ axis where the limit exists
it appears just from the graph that there are holes at the integer values
but not sure what the imaginary part means?

 

[Ackbach]

The W/A plot is missing some important details. I would try plotting $ \lfloor x \rfloor$ and $\lfloor -x \rfloor$ separately, by hand. Then plot the sum $\lfloor x \rfloor+ \lfloor -x \rfloor$ by hand. Pay particular attention to integer values of $x$, and what happens to them.