Note that we do not speak of
\(f\) being continuous or discontinuous at
\(a\text{,}\) as this is not an element of the domain of
\(f\text{.}\)
The limit of
\(f\) at
\(b\) does not exist, since the left limit does not equal the right limit (
5.6). Thus (i) of
Definitionย 6.2 fails, and
\(f\) is not continuous at
\(b\text{.}\) Such a discontinuity is called a
jump discontinuity, as the difference between the left- and right-hand limit values is manifested graphically as a discrete jump in the height of the graph of
\(f\text{.}\)
The limit of \(f\) at \(c\) exists, and is equal to the value of \(f\) at \(c\text{:}\) that is, we have
\begin{equation*}
\lim\limits_{x\to c}f(x)=f(c)=B\text{.}
\end{equation*}
Thus (i) and (ii) of
Definitionย 6.2 are satisfied, and
\(f\) is continuous at
\(c\text{.}\)
The limit of
\(f\) at
\(d\) does not exist, since the right-hand limit does not exist. Thus (i) of
Definitionย 6.2 fails, and
\(f\) is not continuous at
\(d\text{.}\) Such a discontinuity is called an
oscillating discontinuity, as the wild oscillation is the reason why the limit does not exist.
The limit of \(f\) exists at \(e\text{,}\) but is not equal to the value of \(f\) at \(e\text{.}\) In more detail, we have
\begin{equation*}
\lim\limits_{x\to e}f(x)=D< f(e)\text{.}
\end{equation*}
Thus \(f\) is not continuous at \(e\text{.}\) Such a discontinuity is called a removeable discontinuity as a simple redefining of \(f\) at \(e\) (i.e., setting \(f(e)=D\)) would make the discontinuity disappear.