A function is discontinuous at a point when, at that point, one of the conditions required for continuity fails to hold. Intuitively, the graph may exhibit a break, a jump, unbounded growth, or an oscillatory behaviour that prevents the function from approaching a single value.
The notion of discontinuity should not, however, be understood purely in graphical terms. Its classification depends precisely on the behaviour of the limit of the function and, whenever the domain allows the point to be approached from both sides, on the comparison between the left-hand limit and the right-hand limit.
In this article we shall examine rigorously the main types of discontinuity, clarifying the roles of the value \(f(x_0)\), the domain, and the left-hand and right-hand limits. We shall also see why some discontinuities can be removed by altering the value of the function at a single point, whereas others depend on the behaviour of the function in every neighbourhood of the point under consideration.
Contents
- Continuity and discontinuity at a point
- The fundamental role of the limit
- Right-hand and left-hand limits
- Removable discontinuity
- Jump discontinuity
- Infinite discontinuity
- Oscillatory discontinuity
- Traditional classification of discontinuities
- The role of the domain of the function
- Discontinuity and endpoints of the domain
- Graphical interpretation
- Classification examples
- Procedure for studying a discontinuity
- Common mistakes
- Summary
Continuity and discontinuity at a point
Let \( f:A\to\mathbb{R} \) be a real function and let \(x_0\in A\). If \(x_0\) is a limit point of the domain \(A\), the function \(f\) is continuous at \(x_0\) when
\[ \lim_{x\to x_0}f(x)=f(x_0). \]
This equality embodies two fundamental conditions: the limit must exist and be finite, and its value must coincide with the value taken by the function at the point.
If
\[ \lim_{x\to x_0}f(x)=L, \]
continuity therefore requires that
\[ L=f(x_0). \]
If either of these conditions fails, the function is not continuous at \(x_0\).
If \(x_0\) is instead an isolated point of the domain, the function is automatically continuous at \(x_0\). For this reason, in the study of discontinuities attention is focused chiefly on the limit points of the domain, and on points excluded from the domain at which it is nonetheless possible to study the behaviour of the function as \(x\) approaches the point.
Several quite different situations may then arise: the limit may exist without coinciding with \(f(x_0)\); the left-hand limit and the right-hand limit may differ; the function may grow without bound; or it may oscillate indefinitely without approaching any value at all.
The classification of discontinuities arises precisely from the analysis of these different behaviours.
The fundamental role of the limit
To understand discontinuities correctly, it is essential to distinguish the value of the function at the point from the behaviour of the function near the point.
The value
\[ f(x_0) \]
describes what happens exactly at \(x_0\).
The limit
\[ \lim_{x\to x_0}f(x) \]
describes instead the behaviour of \(f(x)\) as \(x\) approaches \(x_0\). The value taken by the function exactly at the point plays no part in the computation of the limit.
Example. Consider
\[ f(x)=\frac{x^2-1}{x-1}, \qquad x\neq1. \]
For \(x\neq1\) we may factorise the numerator:
\[ x^2-1=(x-1)(x+1). \]
Consequently,
\[ f(x)=\frac{(x-1)(x+1)}{x-1}=x+1, \qquad x\neq1. \]
Hence
\[ \lim_{x\to1}f(x) = \lim_{x\to1}(x+1) = 2. \]
The original function is, however, not defined at \(x=1\).
The fact that \(f(1)\) does not exist therefore does not prevent the limit from existing. The limit depends on what happens when \(x\) lies arbitrarily close to \(1\), not on what happens exactly at the point.
This distinction is fundamental to understanding removable discontinuities.
Right-hand and left-hand limits
When the domain allows \(x_0\) to be approached both from values smaller and from values larger than \(x_0\), it is often necessary to study the two behaviours separately.
The left-hand limit is
\[ \lim_{x\to x_0^-}f(x), \]
while the right-hand limit is
\[ \lim_{x\to x_0^+}f(x). \]
When both are meaningful with respect to the domain, the limit
\[ \lim_{x\to x_0}f(x)=L \]
exists and equals \(L\) if and only if the left-hand and right-hand limits both exist and coincide:
\[ \lim_{x\to x_0^-}f(x) = \lim_{x\to x_0^+}f(x) = L. \]
This property allows many discontinuities to be classified.
If, for instance,
\[ \lim_{x\to x_0^-}f(x)=L_1 \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x)=L_2, \]
with
\[ L_1,L_2\in\mathbb{R}, \qquad L_1\neq L_2, \]
then the limit as \(x\to x_0\) does not exist, and a jump discontinuity occurs.
Removable discontinuity
A removable discontinuity occurs when a finite limit
\[ \lim_{x\to x_0}f(x)=L, \qquad L\in\mathbb{R}, \]
exists, but the function does not take the value \(L\) at \(x_0\).
Two situations may arise.
If \(x_0\) belongs to the domain, \(f(x_0)\) is defined, but
\[ f(x_0)\neq L. \]
If, on the other hand, \(x_0\) does not belong to the domain but is a limit point of the domain, then \(f(x_0)\) is not defined. Strictly speaking, the original function cannot be said to be discontinuous at a point that does not belong to its domain. In the study of real functions, however, it is customary to speak in this case too of a removable discontinuity, because the function can be extended continuously by assigning to \(x_0\) the value \(L\).
Example. Consider
\[ f(x)=\frac{x^2-4}{x-2}, \qquad x\neq2. \]
For \(x\neq2\),
\[ f(x) = \frac{(x-2)(x+2)}{x-2} = x+2. \]
Hence
\[ \lim_{x\to2}f(x)=4. \]
Since \(f(2)\) is not defined, the function has a missing point at \(x=2\).
This can be remedied by defining
\[ g(x)= \begin{cases} f(x), & x\neq2,\\ 4, & x=2. \end{cases} \]
This gives
\[ \lim_{x\to2}g(x)=g(2)=4, \]
so that \(g\) is continuous at \(x=2\).
The discontinuity is called removable precisely because it suffices to assign to the point the value of the limit.
Jump discontinuity
A jump discontinuity occurs at \(x_0\) when the left-hand and right-hand limits both exist and are finite, but differ:
\[ \lim_{x\to x_0^-}f(x)=L_1, \qquad \lim_{x\to x_0^+}f(x)=L_2, \]
with
\[ L_1\neq L_2. \]
In this case
\[ \lim_{x\to x_0}f(x) \]
does not exist, because the function approaches two different values according to whether \(x\) tends to \(x_0\) from the left or from the right.
Example. Consider
\[ f(x)= \begin{cases} 1, & x<0,\\ 3, & x\geq0. \end{cases} \]
The left-hand limit is
\[ \lim_{x\to0^-}f(x)=1, \]
while the right-hand limit is
\[ \lim_{x\to0^+}f(x)=3. \]
Since
\[ 1\neq3, \]
the limit as \(x\to0\) does not exist, and the function has a jump discontinuity.
The size of the jump is
\[ |L_2-L_1|. \]
In this example:
\[ |3-1|=2. \]
The value taken by the function exactly at \(x=0\) does not affect the nature of the discontinuity: were \(f(0)\) to be altered, the left-hand limit would remain \(1\) and the right-hand limit would remain \(3\).
A jump discontinuity cannot therefore be removed by altering the value of the function at a single point.
Infinite discontinuity
An infinite discontinuity occurs at \(x_0\) when every left-hand and right-hand limit that is meaningful with respect to the domain exists in \(\overline{\mathbb{R}}\), and at least one of them is equal to \(+\infty\) or \(-\infty\).
For instance, it may happen that
\[ \lim_{x\to x_0^-}f(x)=+\infty \]
or that
\[ \lim_{x\to x_0^+}f(x)=-\infty. \]
In these cases the function does not approach a finite real value and cannot be continuous at the point.
Example. Consider
\[ f(x)=\frac{1}{(x-2)^2}. \]
As \(x\) approaches \(2\), the denominator remains positive and tends to zero. Consequently,
\[ \lim_{x\to2^-}\frac{1}{(x-2)^2}=+\infty \]
and
\[ \lim_{x\to2^+}\frac{1}{(x-2)^2}=+\infty. \]
Hence
\[ \lim_{x\to2}\frac{1}{(x-2)^2}=+\infty. \]
The function therefore has an infinite discontinuity at \(x=2\).
Moreover, the line
\[ x=2 \]
is a vertical asymptote.
In general, for \(x=x_0\) to be a vertical asymptote it suffices that at least one of the right-hand and left-hand limits be infinite.
For example, if
\[ \lim_{x\to x_0^+}f(x)=+\infty, \]
then \(x=x_0\) is a vertical asymptote regardless of what happens to the left of the point.
Oscillatory discontinuity
An oscillatory discontinuity occurs when the failure of a limit to exist is caused by persistent oscillations of the function.
In this case the function does not tend to a finite value or to \(+\infty\) or \(-\infty\), but continues to oscillate as \(x\) approaches the point under consideration.
Example. Consider
\[ f(x)=\sin\left(\frac{1}{x}\right), \qquad x\neq0. \]
As \(x\to0\), the quantity
\[ \frac{1}{x} \]
takes arbitrarily large values in absolute value. Consequently, the sine function continues to oscillate between \(-1\) and \(1\) without tending to a single value.
However small the neighbourhood of \(0\) is taken to be, the function continues to take values arbitrarily close to \(1\), to \(-1\), and to every intermediate value.
Hence
\[ \lim_{x\to0}\sin\left(\frac{1}{x}\right) \]
does not exist.
The non-existence of the limit here does not stem from a jump between two finite values, nor from growth towards \(+\infty\) or \(-\infty\): it is the oscillation of the function itself that prevents the limit from existing.
No alteration of the single value \(f(0)\) could render the function continuous, because the difficulty concerns the behaviour of \(f(x)\) in every neighbourhood of \(0\).
Traditional classification of discontinuities
In school textbooks and in treatises on mathematical analysis, various conventions are found for classifying discontinuities. In particular, the expressions discontinuity of the first kind, of the second kind, and of the third kind are sometimes used.
This terminology is not entirely uniform across authors. For this reason, in a rigorous study of a function it is preferable to describe directly the mathematical behaviour observed.
We shall therefore use the following terminology:
- removable discontinuity, when a finite limit exists but the function does not take the value of the limit at the point;
- jump discontinuity, when the left-hand and right-hand limits both exist, are finite, and differ;
- infinite discontinuity, when all the left-hand and right-hand limits that are meaningful with respect to the domain exist in \(\overline{\mathbb{R}}\), and at least one of them is infinite;
- essential discontinuity, when at least one of the left-hand and right-hand limits that is meaningful with respect to the domain does not exist in \(\overline{\mathbb{R}}\).
This classification has the advantage of describing directly what actually happens to the function, and does not depend on any particular convention used for numbering the various kinds of discontinuity.
The role of the domain of the function
Before classifying a possible discontinuity, it is essential to determine the domain of the function.
A point excluded from the domain does not automatically entail unbounded growth of the function. It is always necessary to study the behaviour of \(f(x)\) as \(x\) approaches that point.
Consider, for instance,
\[ f(x)=\frac{x^2-1}{x-1}. \]
The domain is
\[ \mathbb{R}\setminus\{1\}, \]
yet
\[ \lim_{x\to1}f(x)=2. \]
The function thus admits a continuous extension at \(x=1\) by setting \(f(1)=2\).
Strictly speaking, continuity of a function at a point is defined only when the point belongs to the domain. In the study of functions, however, one commonly speaks of a removable discontinuity at \(x=1\), meaning that the function is undefined at that point but admits a continuous extension there.
The situation is entirely different for
\[ g(x)=\frac{1}{x-1}. \]
In this case
\[ \lim_{x\to1^-}\frac{1}{x-1}=-\infty \]
and
\[ \lim_{x\to1^+}\frac{1}{x-1}=+\infty. \]
The function therefore has an infinite discontinuity at \(x=1\).
Two functions may thus fail to be defined at the same type of point, for example because the denominator vanishes there, and yet display entirely different behaviours.
Discontinuity and endpoints of the domain
A comparison between the left-hand and right-hand limits is necessary only when the domain allows \(x_0\) to be approached from both sides.
If \(x_0\) is instead an endpoint of the domain, continuity is studied by considering the limit from the one side that belongs to the domain.
For example, let
\[ f:[a,b]\to\mathbb{R}. \]
The function is continuous at the left endpoint \(a\) when
\[ \lim_{x\to a^+}f(x)=f(a), \]
while it is continuous at the right endpoint \(b\) when
\[ \lim_{x\to b^-}f(x)=f(b). \]
It would make no sense to require the left-hand limit at \(a\) if the function is defined only for \(x\geq a\), just as it would make no sense to require the right-hand limit at \(b\) if the function is defined only for \(x\leq b\).
The study of continuity must therefore always be consistent with the structure of the domain of the function.
Graphical interpretation
The various types of discontinuity possess different geometric features.
Removable discontinuity
The graph typically shows a missing point. When the point can be reached from both the left and the right, the graph approaches the same ordinate from both sides, but the value of the function is either missing or placed at a different ordinate.
Jump discontinuity
The two branches of the graph approach two different finite ordinates. The vertical distance between these ordinates represents the size of the jump.
Infinite discontinuity
At least one branch of the graph grows or decreases without bound as \(x\) approaches \(x_0\). In this case the line \(x=x_0\) is a vertical asymptote.
Oscillatory discontinuity
The graph continues to oscillate without tending to a single value.
The graphical interpretation is useful for an intuitive grasp of what is happening, but the mathematical classification must be established through the study of limits.
Classification examples
Example 1 (removable discontinuity). Consider
\[ f(x)=\frac{x^2-9}{x-3}. \]
The function is not defined at \(x=3\). For \(x\neq3\),
\[ f(x) = \frac{(x-3)(x+3)}{x-3} = x+3. \]
Hence
\[ \lim_{x\to3}f(x)=6. \]
The limit exists and is finite, while \(f(3)\) is not defined. The discontinuity is removable.
Example 2 (value of the function differing from the limit). Consider
\[ f(x)= \begin{cases} x^2, & x\neq2,\\ 7, & x=2. \end{cases} \]
We have
\[ \lim_{x\to2}f(x)=4, \]
while
\[ f(2)=7. \]
Since
\[ \lim_{x\to2}f(x)\neq f(2), \]
the function is discontinuous at \(x=2\).
The discontinuity is removable, since it would suffice to redefine
\[ f(2)=4. \]
Example 3 (jump discontinuity). Consider
\[ f(x)= \begin{cases} x+1, & x<1,\\ x+3, & x\geq1. \end{cases} \]
The left-hand limit is
\[ \lim_{x\to1^-}f(x)=2, \]
while the right-hand limit is
\[ \lim_{x\to1^+}f(x)=4. \]
Since
\[ 2\neq4, \]
the limit as \(x\to1\) does not exist, and the function has a jump discontinuity.
Example 4 (infinite discontinuity). Consider
\[ f(x)=\frac{1}{x-1}. \]
As \(x\to1^-\), the denominator tends to zero through negative values:
\[ \lim_{x\to1^-}\frac{1}{x-1}=-\infty. \]
As \(x\to1^+\), the denominator tends to zero through positive values:
\[ \lim_{x\to1^+}\frac{1}{x-1}=+\infty. \]
The function therefore has an infinite discontinuity at \(x=1\), and the line
\[ x=1 \]
is a vertical asymptote.
Example 5 (oscillatory discontinuity). Consider
\[ f(x)=\cos\left(\frac{1}{x}\right), \qquad x\neq0. \]
As \(x\to0\), the argument \(1/x\) takes arbitrarily large values in absolute value, and the cosine keeps oscillating between \(-1\) and \(1\).
Hence
\[ \lim_{x\to0}\cos\left(\frac{1}{x}\right) \]
does not exist.
The function therefore has an oscillatory discontinuity at \(x=0\).
Procedure for studying a discontinuity
To determine correctly the nature of a possible discontinuity at \(x_0\), it is convenient to proceed systematically.
Determine the domain
One must first establish whether \(x_0\) belongs to the domain and from which sides the point may be approached.
Study the left-hand and right-hand limits
When both are meaningful with respect to the domain, one computes
\[ \lim_{x\to x_0^-}f(x) \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x). \]
If the domain allows the point to be approached from only one side, only the corresponding limit is considered.
Check whether a finite limit exists
If the left-hand and right-hand limits are both finite and coincide,
\[ \lim_{x\to x_0^-}f(x) = \lim_{x\to x_0^+}f(x) = L, \]
then
\[ \lim_{x\to x_0}f(x)=L. \]
If \(x_0\) belongs to the domain, one must compare \(L\) with \(f(x_0)\).
If
\[ f(x_0)=L, \]
the function is continuous at \(x_0\).
If instead
\[ f(x_0)\neq L, \]
the discontinuity is removable.
If \(x_0\) does not belong to the domain but the finite limit \(L\) exists, the function can be extended continuously by assigning to the point the value \(L\).
Check for the presence of a jump
If the left-hand and right-hand limits both exist and are finite, but differ, the discontinuity is a jump discontinuity.
Check for infinite behaviour
If every left-hand and right-hand limit that is meaningful with respect to the domain exists in \(\overline{\mathbb{R}}\), and at least one of them equals \(+\infty\) or \(-\infty\), the function has an infinite discontinuity at \(x_0\).
Check for oscillatory behaviour or other failure of the limit
If a left-hand or right-hand limit that is meaningful with respect to the domain fails to exist because the function continues to oscillate without tending to a finite value or to \(+\infty\) or \(-\infty\), the function has an oscillatory discontinuity.
This procedure avoids classifying a discontinuity merely by inspecting the formula of the function or the mere fact that a given point is excluded from the domain.
Common mistakes
Confusing the limit with the value of the function
The fact that \(f(x_0)\) is not defined does not imply that
\[ \lim_{x\to x_0}f(x) \]
fails to exist.
The limit describes the behaviour of the function as \(x\) approaches \(x_0\), not what happens exactly at the point.
Automatically regarding a discontinuity caused by a vanishing denominator as infinite
If the denominator of a rational function vanishes at \(x_0\), one must check whether the numerator and the denominator share common factors.
For instance,
\[ \frac{x^2-1}{x-1} \]
has a removable discontinuity at \(x=1\), whereas
\[ \frac{1}{x-1} \]
has an infinite discontinuity.
Looking only at \(f(x_0)\)
Knowing the value of the function at the point is not sufficient to establish continuity.
It may happen, for instance, that \(f(x_0)\) is perfectly well defined, yet the left-hand and right-hand limits differ. In that case the function remains discontinuous regardless of the value assigned to \(f(x_0)\).
Failing to distinguish the right-hand limit from the left-hand limit
At points where the formula of the function changes, where the denominator vanishes, or where the behaviour may differ on the two sides, it is often essential to study separately
\[ \lim_{x\to x_0^-}f(x) \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x). \]
Concluding that the limit does not exist when both the left-hand and right-hand limits equal \(+\infty\)
If
\[ \lim_{x\to x_0^-}f(x)=+\infty \]
and
\[ \lim_{x\to x_0^+}f(x)=+\infty, \]
one correctly writes
\[ \lim_{x\to x_0}f(x)=+\infty. \]
No finite real limit exists, but an infinite limit does.
Classifying two differing infinite behaviours as a jump
If
\[ \lim_{x\to x_0^-}f(x)=-\infty \qquad\text{and}\qquad \lim_{x\to x_0^+}f(x)=+\infty, \]
this is not a jump discontinuity.
A jump discontinuity requires, in fact, that the left-hand and right-hand limits both be finite.
In this case the discontinuity is infinite.
Summary
The classification of a discontinuity depends on the behaviour of the function as \(x\) approaches the point in question and, subsequently, on the comparison with the value, if any, taken by the function at that point.
| Behaviour at \(x_0\) | Conclusion |
|---|---|
| \(\displaystyle\lim_{x\to x_0}f(x)=f(x_0)=L\) | Continuous function |
| \(\displaystyle\lim_{x\to x_0}f(x)=L\in\mathbb{R}\), but \(f(x_0)\) is not defined or \(f(x_0)\neq L\) | Removable discontinuity |
| \(\displaystyle\lim_{x\to x_0^-}f(x)=L_1\), \(\displaystyle\lim_{x\to x_0^+}f(x)=L_2\), with \(L_1,L_2\in\mathbb{R}\) and \(L_1\neq L_2\) | Jump discontinuity |
| All the left-hand and right-hand limits that are meaningful with respect to the domain exist in \(\overline{\mathbb{R}}\), and at least one of them equals \(+\infty\) or \(-\infty\) | Infinite discontinuity |
| At least one of the left-hand and right-hand limits that is meaningful with respect to the domain does not exist in \(\overline{\mathbb{R}}\) | Essential discontinuity; oscillatory when the non-existence is due to persistent oscillations |
The study of discontinuities is thus, first and foremost, a study of the local behaviour of the function.
If a finite limit exists, comparison with \(f(x_0)\) determines whether the function is continuous or has a removable discontinuity. If the left-hand and right-hand limits are finite but differ, there is a jump discontinuity. If at least one of the left-hand and right-hand limits is infinite, there is an infinite discontinuity. If the failure of a left-hand or right-hand limit to exist is caused by persistent oscillations, there is an oscillatory discontinuity.
A systematic comparison between the domain, the left-hand limit, the right-hand limit, and the value of the function thus makes it possible to identify and classify rigorously the behaviour of the function at every significant point of the domain, or in its vicinity.