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A limit specifies the value approached by a function as its input approaches a point through other points of its domain. For $f:D\subseteq \mathbb{R}\to \mathbb{R}$ and an accumulation point $a$ of $D$, the statement ${\lim }_{x\to a}f(x)=L$ meansThe exclusion of $x=a$ matters: $f(a)$ can differ from $L$, or be undefined. A continuous function is one whose limit agrees with its point value. Requiring $a$ to be an accumulation point avoids a vacuous condition at an isolated input.^[ref]

For $f(x)=(x^2-1)/(x-1)$ on $\mathbb{R}\setminus \{1\}$, cancellation gives $f(x)=x+1$, so the limit at $1$ is $2$ despite the missing point value. A removable discontinuity can be repaired by assigning that value. In contrast, $\sin (1/x)$ has no limit at zero: sequences approaching zero can make its values identically $1$ or identically $-1$. This uses the sequential criterion for a function limit.