Is it necessary that if a limit exists at a point it should be also defined at that point?

Question

Say there exists a limit $\lim_{x \to x_0}f(x) = L$. Is it necessary that $f$ be defined at the point $x_0$ itself?
Well, what I think of it is that it's OK to be undefined at that point because I guess that won't cease the limit of that function to exist there would it?

Answer 1

"No", in a particularly strong sense: The very reason limits came into mathematics was to extract numerical values from difference quotients $$ \frac{f(x) - f(x_{0})}{x - x_{0}} $$ (which are, of course, algebraically indeterminate at $x_{0}$), in the limit as $x \to x_{0}$.

The formal definition of $L = \lim(f, x_{0})$ ("For every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - x_{0}| < \delta$, then $|f(x) - L| < \varepsilon$") explicitly avoids evaluation of $f$ at $x_{0}$, precisely because $f$ might be undefined at $x_{0}$.

Answer 2

This is not at all necessary. For example: $$ \lim_{x\to 0} \frac {x^2}x =0, $$ but $\left.\dfrac{x^2}{x}\right\vert_{x=0}$ is undefined.

If we do have that $f$ is defined at $a$ and $\lim\limits_{x\to a}f(x)=f(a)$, then $f$ is called continuous at $x=a$.

Answer 3

$\lim_{x\to x_0} f(x)=L$ for a $f:D\to\mathbb R$ is defined as

For all sequences $(a_n)$ with $\lim_{n\to\infty} a_n = x_0$ we find $\lim_{n\to\infty} f(a_n) = L$.

To guarantee that $L$ is uniquely defined we need at least one sequence $(a_n)$ of the domain such as $\lim_{n\to\infty} a_n = x_0$. Therefore $x_0$ need to be an accumulation point of the domain $D$.

Conclusion: $x_0$ does not need to be in the domain of $f$ but it must be an accumulation point of the domain.