Consider the function $$ f(x):=x\text{ mod }2\pi,~~x\in [0,2\pi). $$
Then obviously, $f$ is not defined for $x=2\pi$ but $\lim_{x^-\to 2\pi}f(x)=0$.
I am not sure whether $x=2\pi$ is a removable discontinuity.
I think this is the case of $\lim_{x\to 2\pi}f(x)$ exists but I am not sure what $\lim_{x^+\to 2\pi}f(x)$ is (how can $x^+\to 2\pi$ make any sense if $x\in [0,2\pi)$?) resp. if it coincides with $\lim_{x^-\to 2\pi}f(x)=0$.
The question is 'ill-posed' because discontinuity points are generally assumed to be in the interior of the domain of continuity. Being at the frontier, your point cannot be 'eliminated' by defining the value of the function there as the limit of the function there, simply because there's no two-sided limit at that point. You surely can readapt the canonical definitions to your needs but I think you'll hardly come up with something useful (you would be limited to this kind of context, where your function has whole intervals removed from its domain). Also definitions are just names used to refer to commonly used things, but that doesn't mean that your point can't be interesting anyway.