A real function $f$ is said to be continuous if it is continuous at every point in the domain of $f$.
Suppose $f$ is a function defined on $[a,b]$, then for $f$ to be continuous, it needs to be continuous at every point in $[a, b]$ including the end points $a$ and $b$.
Continuity of $f$ at $a$$\implies \lim_\limits{x\to{a^+}}f(x)=f(a)$
continuity of $f$ at $b$$\implies \lim_\limits{x\to{b^-}}f(x)=f(b)$
Observe that $\lim_\limits{x\to{a^+}}f(x)=f(a)$ and $\lim_\limits{x\to{b^-}}f(x)=f(b)$ do not make sense. As a consequence of this definition, if $f$ is defined only at one point, it is continuous there, i.e., if the domain of $f$ is a singleton, $f$ is a continuous function.
First two sentences seems fine.
But what does the rest really means ?. What it means to say that "$\lim_\limits{x\to{a^+}}f(x)=f(a)$ and $\lim_\limits{x\to{b^-}}f(x)=f(b)$ do not make sense" ?
It seems you may have mixed up $+$ and $-$ when it comes to 'make no sense'.
if $f$ is (only) defined on $[a,b]$, then it is not defined for $x<a$ or $x>b$.
$\lim_{x\rightarrow a_-}$ means, however, $\lim_{x\rightarrow a, x< a}$, which does not make sense in this case because $f$ is not defined on the set in question. Similarly for $b_+$