Let $f:R \to R$ be $f(x)=\lfloor x \rfloor+ \lfloor -x \rfloor$ (floor function)
Prove or disprove: The limit $\lim_{x \to x_0}f(x)$ exists for every $x_0\in R$ and define what types of point discontinuities the function has, if any.
1.) if the limit exists we need to prove that both one-sided limits exist and are equal to each other.
$\lim_{x\to x_0^+}\lfloor x \rfloor+ \lfloor -x \rfloor$ = $\lfloor x_0^+ \rfloor+ \lfloor -x_0^+ \rfloor$= $x_0 - x_0 =0$
Similarly:
$\lim_{x\to {x_0-}}f(x)=0$
But this seems to only works for $x_0\gt0$
Picking $x_0=-4.3$ we get that f(x)=$\lfloor -4.3 \rfloor+ \lfloor 4.3 \rfloor=-5+4=-1$
Checking the one sided limits, we get that indeed for $x_0\lt 0$ exist and are equal to each other.
Does this make sense?
And for Discontinuities, it makes sense for them to be at 0 since picking something a little bit smaller than zero will give us -2, picking something a little bigger will give us 0. Therefore the discontinuity will be that the one-sided limits are not equal to each other (not sure of name.)
Would appreciate a quick look over to see if I'm right.