Q.Discuss the continuity of the following function at the point(s) or on the interval indicated against it: $f(x) = \lfloor x+1 \rfloor$ for $x \in [-2,2)$, where $\lfloor \, \rfloor$ is the greatest integer function.

$f(x)=\lfloor x+1\rfloor$ for $x\in[-2,2)$. As $x$ ranges over $[-2,2)$, $x+1$ ranges over $[-1,3)$, and $\lfloor x+1\rfloor$ jumps wherever $x+1$ is an integer, i.e. at $x=-1,0,1,2$; but $x=2$ is excluded from the domain, so the relevant jump points inside $[-2,2)$ are $x=-1,0,1$.

At each such point $n\in\{-1,0,1\}$: $f(n)=\lfloor n+1\rfloor=n+1$ (right-continuous value), while $\displaystyle\lim_{x\to n^-} f(x)=n$ (the value just below). Since $n\ne n+1$, both one-sided limits exist (as $n$ and $n+1$) but differ — a jump discontinuity of size $1$ at each of $x=-1,0,1$. …

Track where the inner expression $x+1$ crosses an integer, since that is exactly where the floor function steps; then check which of …

Wrongly including the domain's own left endpoint $x=-2$ among the discontinuities — continuity there only requires the one-sided (right) limit to match $f(-2)$, wh …