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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of discontinuities | 1/4 | https://en.wikipedia.org/wiki/Classification_of_discontinuities | reference | science, encyclopedia | 2026-05-05T09:08:17.602531+00:00 | kb-cron |
While continuous functions are important in mathematics, not all functions are continuous. If a function is not continuous at a limit point (also called an "accumulation point" or "cluster point") of its domain, it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.
== Classification == For each of the following, consider a real valued function
f
{\displaystyle f}
of a real variable
x
,
{\displaystyle x,}
defined in a neighborhood of the point
x
0
{\displaystyle x_{0}}
at which
f
{\displaystyle f}
is discontinuous.
=== Removable discontinuity ===
Consider the piecewise function
f
(
x
)
=
{
x
2
for
x
<
1
0
for
x
=
1
2
−
x
for
x
>
1
{\displaystyle f(x)={\begin{cases}x^{2}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\2-x&{\text{ for }}x>1\end{cases}}}
The point
x
0
=
1
{\displaystyle x_{0}=1}
is a removable discontinuity. For this kind of discontinuity: The one-sided limit from the negative direction:
L
−
=
lim
x
→
x
0
−
f
(
x
)
{\displaystyle L^{-}=\lim _{x\to x_{0}^{-}}f(x)}
and the one-sided limit from the positive direction:
L
+
=
lim
x
→
x
0
+
f
(
x
)
{\displaystyle L^{+}=\lim _{x\to x_{0}^{+}}f(x)}
at
x
0
{\displaystyle x_{0}}
both exist, are finite, and are equal to
L
=
L
−
=
L
+
.
{\displaystyle L=L^{-}=L^{+}.}
In other words, since the two one-sided limits exist and are equal, the limit
L
{\displaystyle L}
of
f
(
x
)
{\displaystyle f(x)}
as
x
{\displaystyle x}
approaches
x
0
{\displaystyle x_{0}}
exists and is equal to this same value. If the actual value of
f
(
x
0
)
{\displaystyle f\left(x_{0}\right)}
is not equal to
L
,
{\displaystyle L,}
then
x
0
{\displaystyle x_{0}}
is called a removable discontinuity. This discontinuity can be removed to make
f
{\displaystyle f}
continuous at
x
0
,
{\displaystyle x_{0},}
or more precisely, the function
g
(
x
)
=
{
f
(
x
)
x
≠
x
0
L
x
=
x
0
{\displaystyle g(x)={\begin{cases}f(x)&x\neq x_{0}\\L&x=x_{0}\end{cases}}}
is continuous at
x
=
x
0
.
{\displaystyle x=x_{0}.}
The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point
x
0
.
{\displaystyle x_{0}.}
This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain.
=== Jump discontinuity ===
Consider the function
f
(
x
)
=
{
x
2
for
x
<
1
0
for
x
=
1
2
−
(
x
−
1
)
2
for
x
>
1
{\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-(x-1)^{2}&{\mbox{ for }}x>1\end{cases}}}
Then, the point
x
0
=
1
{\displaystyle x_{0}=1}
is a jump discontinuity. In this case, a single limit does not exist because the one-sided limits,
L
−
{\displaystyle L^{-}}
and
L
+
{\displaystyle L^{+}}
exist and are finite, but are not equal: since,
L
−
≠
L
+
,
{\displaystyle L^{-}\neq L^{+},}
the limit
L
{\displaystyle L}
does not exist. Then,
x
0
{\displaystyle x_{0}}
is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function
f
{\displaystyle f}
may have any value at
x
0
.
{\displaystyle x_{0}.}
=== Essential discontinuity ===
For an essential discontinuity, at least one of the two one-sided limits does not exist in
R
{\displaystyle \mathbb {R} }
. (Notice that one or both one-sided limits can be
±
∞
{\displaystyle \pm \infty }
). Consider the function
f
(
x
)
=
{
sin
5
x
−
1
for
x
<
1
0
for
x
=
1
1
x
−
1
for
x
>
1.
{\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\{\frac {1}{x-1}}&{\text{ for }}x>1.\end{cases}}}
Then, the point
x
0
=
1
{\displaystyle x_{0}=1}
is an essential discontinuity. In this example, both
L
−
{\displaystyle L^{-}}
and
L
+
{\displaystyle L^{+}}
do not exist in
R
{\displaystyle \mathbb {R} }
, thus satisfying the condition of essential discontinuity. So
x
0
{\displaystyle x_{0}}
is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from an essential singularity, which is often used when studying functions of complex variables).
== Counting discontinuities of a function == Supposing that
f
{\displaystyle f}
is a function defined on an interval
I
⊆
R
,
{\displaystyle I\subseteq \mathbb {R} ,}
we will denote by
D
{\displaystyle D}
the set of all discontinuities of
f
{\displaystyle f}
on
I
.
{\displaystyle I.}
By
R
{\displaystyle R}
we will mean the set of all
x
0
∈
I
{\displaystyle x_{0}\in I}
such that
f
{\displaystyle f}
has a removable discontinuity at
x
0
.
{\displaystyle x_{0}.}
Analogously by
J
{\displaystyle J}
we denote the set constituted by all
x
0
∈
I
{\displaystyle x_{0}\in I}
such that
f
{\displaystyle f}
has a jump discontinuity at
x
0
.
{\displaystyle x_{0}.}
The set of all
x
0
∈
I
{\displaystyle x_{0}\in I}
such that
f
{\displaystyle f}
has an essential discontinuity at
x
0
{\displaystyle x_{0}}
will be denoted by
E
.
{\displaystyle E.}
Of course then
D
=
R
∪
J
∪
E
.
{\displaystyle D=R\cup J\cup E.}
The two following properties of the set
D
{\displaystyle D}
are relevant in the literature.