Definition: A function $f$ is continuous at a number $a$ if $\displaystyle \lim_{x \rightarrow a} f(x) = f(a)$.

Remember that $\displaystyle{\lim_{x \to a} f(x)}$ describes both what is
happening when $x$ is slightly less than $a$ and what is happening when $x$
is slightly greater than $a$. A function is continuous at $a$
if the lead-up to $a$
(that is, $\displaystyle{\lim_{x \to a^-} f(x)}$), the value at $a$ (that is,
$f(a)$)
and the run-out from $a$ (that is, $\displaystyle{\lim_{x \to a^+} f(x)}$)
all match.
If any of these quantities is different, or if any of them fails to exist,
then we say that $f(x)$ is discontinuous at $x=a$, or that $f(x)$ has
a discontinuity at $x=a$.

Definition: A function $f$ is
continuous from the right at a number $a$ if
$\displaystyle \lim_{x \rightarrow a^+} f(x) = f(a)$ and is
continuous from the left if
$\displaystyle \lim_{x \rightarrow a^-} f(x) = f(a)$. A function is
continuous on an interval $I$ if it is continuous at each interior point
of $I$, is continuous from the right at the left endpoint (if $I$ has one),
and is continuous from the left at the right endpoint (if $I$ has one).