A question about continuity at a point

Question

I have a question about continuity of a function defined in terms of limits. For simplicity consider only a function $f: \mathbb{R} \rightarrow \mathbb{R}$. I have seen a sufficient condition for continuity at a point $x_{*}$ to be written as

$\displaystyle \lim_{x\rightarrow x^{-}_{*}}f(x) = \lim_{x\rightarrow x_{*}^{+}}f(x). \tag{1}$

My question is that if $f$ has a pointwise discontinuity at $x_{*}$, but has the same value around $x_{*}$ (for example)

$f(x) = \begin{cases}1, & x \neq 1 \\ 0, & x = 1\end{cases} $

then as I understand it the left hand limit $\displaystyle \lim_{x\rightarrow 1^{-}}f(x) = 1 = \lim_{x\rightarrow 1^{+}}f(x)$ which would imply that $f$ is continuous at $1$, which it clearly isn't. Am I missing something here or is my definition of pointwise continuity (1) not sufficient?

Answer 1

Indeed the equality of the one-sided limits does not guarantee continuity, only that the full limit $\lim_{x \to x_*} f(x)$ exists. You need

$$\lim_{x\rightarrow x^{-}_{*}}f(x) = \lim_{x\rightarrow x_{*}^{+}}f(x) = f(x_*)$$

for continuity.