Continuity of $\sqrt{x}$

Question

If a function is not defined on a set $I \subset \mathbb{R}$ could we say that this function is not continuous on $I$ ?

For example the function $\sqrt{x}$ is not defined on $\mathbb{R}^{-}$ but can we say that this function is not continuous on $\mathbb{R}^{-}$?

I am asking this question because by definition a function is continuous on $a$ if $\lim_{x \rightarrow a} f(x) \in \mathbb{R}$, but if $a \in I$ then $\lim_{x \rightarrow a} f(x)$ doesn't exist and hence is not in $\mathbb{R}$ and thus the function is not continuous.

Answer 1

Technically speaking, the sentence "the function $f$ is continuous on $I$" implicitly means that $f$ is defined on $I$. Therefore, if $f$ is not defined on $I$, it is technically correct to say that $f$ is not continuous on $I$.

However, it is also much better to err on the safe side so just say "it is not continuous there because it is not defined".

Answer 2

The question is meaningless. It only makes sense to ask if a function $f\colon A\longrightarrow B$ is continuous if $A$ is part of the domain of $f$.

Answer 3

Definition:

A function $f:\Bbb R\to \Bbb R$ is continuous at $c\in\Bbb R$ if $\forall ε > 0,\exists δ > 0,\,|x - c| < δ \implies |f (x) - f (c)| < ε$

Now tell me, if $f(c)$ is undefined, can $f(x)-f(c)$ be defined?