One-sided limits of the rounding function at a half-integer

For $x$ slightly less than $2.5$, such as $2.4, 2.49, 2.499$, the nearest integer is always $2$, since these values are closer to $2$ than to $3$. So $\displaystyle\lim_{x \to 2.5^-} k(x) = 2$.

For $x$ slightly greater than $2.5$, such as $2.6, 2.51, 2.501$, the nearest integer is always $3$, since these values are closer to $3$ than to $2$. So $\displaystyle\lim_{x \to 2.5^+} k(x) = 3$. Since the left-hand limit ($2$) and right-hand limit ($3$) are not equal, the two-sided limit $\displaystyle\lim_{x \to 2.5} k(x)$ does not exist.