Removable and essential singularities
A singularity of $f$ at $a$ is removable if $\displaystyle \lim_{x\to a}f(x)$ exists. If the limit exists, then we can extend the domain of $f$ to include $a$ and make $f$ continuous at $a$ by setting $\displaystyle f(a) = \lim_{x\to a}f(x)$, thereby removing the singularity. On the other hand, if $\displaystyle \lim_{x\to a}f(x)$ does not exist, then no possible value for $f(a)$ could make $f$ continuous at $a$, so, in that case, we say that the singularity at $a$ is essential.
Removable singularity point at $a$: $\displaystyle \lim_{x\to a}f(x)= 1$
Defining $f(a) = 1$ results in a continuous function.
Defining $f(a) = 1$ results in a continuous function.