Firstly continuity at end-points is traditionally defined via one sided limits.
$f$ is continuous at $-6$ in the sense that $\lim_{x\to 6^+}f(x)$ exists and is $f(6)$
More formally: $f$ is continuous in $[a,b]$ if it is continuous in its interior and $\lim_{x\to a^+}f(x)=f(a)$ and $\lim_{x\to b^-}f(x)=f(b)$
Secondly the limit $\lim_{x\to a^-}f(x)$ may not be defined (in the sense that $x$ can't approach $a$ from the left) but the limit $\lim_{x\to a}f(x)$ does exist. In fact:
If $f:X\to \mathbb{R}$ and $a$ is an accumulation point of $X$ only from the right (*) and $\lim_{x\to a^+}f(x)=L$ then $\lim_{x\to a}f(x)=L$
(*): That is $\forall \delta>0$, $(a,a+\delta)\cap X\neq \emptyset$ (accumation from the right) while $\exists\delta>0$, $(a-\delta,a)\cap X=\emptyset$ (non accumulation from the left)
Note: The statement: the limit exists iff the one sided lmits exist and are equal is not correct. The correct statement is the following:
The limit exists at point that is a limit point from the left and from the right iff the one sided lmits exist and are equal
The limit exists at point that is a limit point only from the left iff the left one sided lmits exists.
The limit exists at point that is a limit point only from the right iff the right one sided lmits exists.