# Discontinuities of Real Functions
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https://algebrica.org/discontinuities-of-real-functions/
Fetched from algebrica.org post 24269; source modified 2026-03-01T13:01:39.
## Introduction
Continuity is a property of a [function](../functions.md) in which small variations in the input result in correspondingly small variations in the output within the neighbourhood of a given point. If this local stability does not hold, the function is considered discontinuous. Discontinuities are typically classified into three distinct types:
- A removable discontinuity occurs when the limit exists and is finite, but the function is either undefined at the point or its value does not equal the limit.
- A jump discontinuity is present when both the left-hand and right-hand limits exist and are finite, but these limits are not equal.
- An infinite discontinuity occurs when at least one of the one-sided limits is infinite, causing the function to diverge near the point rather than approach a finite value.
A discontinuity at \\(x\_0\\) can occur in exactly one of the three mutually exclusive ways described above. A point cannot simultaneously exhibit more than one type of discontinuity.
###### Each of these types will be examined in detail in the following sections. In this discussion, \\( f \\) denotes a real-valued function, and \\( x\_0 \\) represents a point in its [domain](../determining-the-domain-of-a-function.md) or a point at which the function may fail to be defined.
## Recall of continuity
A function \\( f \\) is [continuous](../continuous-functions.md) at \\( x\_0 \\) if the limit as \\( x \\) approaches \\( x\_0 \\) exists, is finite, and coincides with the value of the function at that point. This condition is expressed by the following [limit](../limits.md):
\\[
\\lim\_{x \\to x\_0} f(x) = f(x\_0)
\\]
[Polynomials](../polynomials.md) constitute a fundamental class of elementary continuous functions. These functions represent smooth curves in the plane and exhibit no points of discontinuity. Below is the graph of the quadratic function \\( x^2 + 2x + 1 \\), which represents a [parabola](../parabola.md):
A discontinuity at \\( x\_0 \\) arises whenever this equality does not hold, and the specific way in which the condition fails determines the type of discontinuity.
###### Intuitively, a function is continuous if its graph can be drawn in the plane without any interruptions, breaks, or sudden jumps.
## Removable discontinuity
A removable discontinuity arises when a function possesses a well-defined finite limit at \\( x\_0 \\), yet the function’s value at that point is either undefined or does not coincide with the limit. Formally, a function \\( f \\) has a removable discontinuity at \\( x\_0 \\) if the following limit exists and is finite:
\\[
\\lim\_{x \\to x\_0} f(x) = \\ell \\in \\mathbb{R}
\\]
Moreover, at least one of the following conditions is satisfied:
- \\( f(x\_0) \\) is undefined.
- \\( f(x\_0) \\neq \\ell \\).
In such cases, the discontinuity may be removed by redefining the function at a single point as follows:
\\[
g(x) = \\begin{cases} \\ell & \\text{if } x = x\_0 \\[6pt] f(x) & \\text{if } x \\ne x\_0 \\end{cases}
\\]
With this definition, the function \\( g \\) becomes continuous at \\( x\_0 \\). The term “removable” refers to the fact that the discontinuity can be resolved in this manner.
###### Removable discontinuities typically occur in rational functions containing cancellable factors, resulting in a hole in the graph. They can also be present in piecewise-defined functions or in functions where the value at a single point has been modified, provided the limit at that point exists and is finite.
## Example 1
Consider the function defined by the following [rational](../rational-functions.md) expression, which is undefined at \\( x = 1 \\):
\\[
f(x) = \\frac{x^2 - 1}{x - 1}
\\]
[Factoring](../factoring-ac-method.md) the numerator demonstrates that the expression simplifies for all values of \\( x \\) except \\(1\\) since \\(x=1\\) would cancel the denominator and make the function undefined.
\\[
x^2 - 1 = (x - 1)(x + 1)
\\]
For all \\( x \\neq 1 \\), the function is equivalent to a linear function:
\\[
f(x) = x + 1
\\]
Although the function is undefined at \\( x = 1 \\), the limit as \\( x \\) approaches \\(1\\) exists and is finite:
\\[
\\lim\_{x \\to 1} \\frac{x^2 - 1}{x - 1} = 2
\\]
This demonstrates that \\( x = 1 \\) is a removable discontinuity, as the graph corresponds to the straight line \\( y = x + 1 \\) with a single missing point at\\( (1,2) .\\)
Redefining the function at that point by assigning it the value of the limit eliminates the discontinuity:
\\[
g(x) = \\begin{cases} 2 & \\text{if } x = 1 \\[6pt] f(x) & \\text{if } x \\ne 1 \\end{cases}
\\]
###### With this modification, the function is continuous at \\( x = 1 \\).
## Jump discontinuity
A jump discontinuity arises when both the left-hand and right-hand limits at \\( x\_0 \\) exist and are finite, yet these limits are not equal. Formally, \\( f \\) has a jump discontinuity at \\( x\_0 \\) if:
\\[
\\begin{align} \\lim\_{x \\to x\_0^-} f(x) &= \\ell\_1 \\in \\mathbb{R} \\[6pt] \\lim\_{x \\to x\_0^+} f(x) &= \\ell\_2 \\in \\mathbb{R} \\[6pt] \\ell\_1 &\\neq \\ell\_2 \\end{align}
\\]
In this case, the limit \\( \\lim\_{x \\to x\_0} f(x) \\) does not exist. The function approaches two distinct finite values depending on the direction of approach. Unlike a removable discontinuity, this type cannot be resolved by redefining the function at a single point, as the discrepancy is inherent to the local behavior.
## Example 2
To analyse the jump discontinuity, consider the following simple function, which exhibits a discontinuity at the point \\(x = 1.\\)
\\[
f(x) = \\begin{cases} 0 & \\text{if } x < 1 \\[6pt] 2 & \\text{if } x \\ge 1 \\end{cases}
\\]
For values of \\(x\\) approaching \\(1\\) from the left, the function remains constant at \\(0\\). Therefore:
\\[
\\lim\_{x \\to 1^-} f(x) = 0
\\]
For values of \\( x \\) approaching 1 from the right, the function remains constantly equal to \\(2\\), and therefore the limit is:
\\[
\\lim\_{x \\to 1^+} f(x) = 2
\\]
Both one-sided limits exist and are finite but they are not equal. Since \\( 0 \\neq 2 \\), it follows that the two one-sided limits do not coincide, and consequently, the limit \\(\\lim\_{x \\to 1} f(x)\\) does not exist. The graph of the function shows a vertical jump at \\(x = 1\\), transitioning from \\(0\\) to \\(2\\).
This discontinuity cannot be removed by redefining the function at \\(x = 1\\), as the difference between the two limiting values indicates a break in the local behaviour of the function.
## Infinite discontinuity
An infinite discontinuity occurs when a function diverges as \\( x \\) approaches \\( x\_0 \\), with at least one of the one-sided limits being infinite. Formally, a function \\( f \\) exhibits an infinite discontinuity at \\( x\_0 \\) if at least one of the following conditions is satisfied:
\\[
\\begin{align} \\lim\_{x \\to x\_0^-} f(x) &= \\pm \\infty \\[6pt] \\lim\_{x \\to x\_0^+} f(x) &= \\pm \\infty \\end{align}
\\]
In such cases, the function does not approach any finite value as \\( x \\) nears \\( x\_0 \\). The graph typically displays a vertical [asymptote](../asymptotes.md). This discontinuity reflects unbounded growth rather than a finite discontinuity.
## Example 3
For example, consider the following function:
\\[
f(x) = \\frac{1}{x - 2}
\\]
The behaviour of this function near \\( x\_0 = 2 \\) is analysed as follows. As \\( x \\to 2^- \\), the denominator \\( x - 2 \\) becomes negative and approaches zero, causing the function to decrease without bound. Therefore we have:
\\[
\\lim\_{x \\to 2^-} f(x) = -\\infty
\\]
As \\( x \\to 2^+ \\), the denominator is positive and approaches zero, which causes the function to increase without bound. The limit is:
\\[
\\lim\_{x \\to 2^+} f(x) = +\\infty
\\]
At least one of the one-sided limits is infinite, and they diverge with opposite signs. Consequently, the function exhibits an infinite discontinuity at \\( x = 2 \\). The graph displays a [vertical asymptote](../asymptotes.md) at the line \\( x = 2 \\), and this divergence indicates unbounded growth rather than a finite jump or a removable discontinuity.
## Discontinuity, continuity and differentiability
It is instructive to establish a precise link between the notions of discontinuity and [differentiability](../derivatives.md). We know that if a function f is differentiable at a point \\(x\_0\\), it must also be [continuous](../continuous-functions.md) at that point. The existence of the derivative ensures that the function satisfies the condition of continuity:
\\[
f’(x\_0) = \\lim\_{x \\to x\_0} \\frac{f(x) - f(x\_0)}{x - x\_0}
\\]
\\[
\\lim\_{x \\to x\_0} f(x) = f(x\_0)
\\]
Therefore, if a function exhibits a discontinuity of the type just described at \\( x\_0 \\), meaning the limit does not exist or does not equal the function’s value the derivative at that point does not exist.
However, the converse is not true. A function can be continuous at \\(x\_0\\) yet not differentiable there. This situation arises when the one-sided derivatives exist but differ, when at least one is infinite, or when one of the limits diverges.
\\[
\\lim\_{x \\to x\_0^-} \\frac{f(x) - f(x\_0)}{x - x\_0} \\neq \\lim\_{x \\to x\_0^+} \\frac{f(x) - f(x\_0)}{x - x\_0}
\\]
A common example is the [absolute value function](../absolute-value-function.md), which is continuous at \\(x = 0\\) but not differentiable there, resulting in a corner on its graph. In summary, every discontinuity implies non-differentiability, whereas not every point of non-differentiability is associated with a discontinuity.
## A particular case: essential discontinuity
An additional category, known as essential discontinuity, is sometimes recognised but not universally adopted as a formal classification. This type arises when the limit does not exist and cannot be described as infinite. Unlike a jump discontinuity, where both one-sided limits exist but are unequal, or an infinite discontinuity, where the function diverges in a particular direction, an essential discontinuity reflects fundamentally irregular behaviour that cannot be reduced to simpler forms.
A classic example is the following function, which exhibits an essential discontinuity at \\(x = 0\\):
\\[
f(x) = \\sin\\left(\\frac{1}{x}\\right)
\\]
As \\(x\\) approaches \\(0\\), the argument \\(1/x\\) grows without bound, causing the function to oscillate between \\(-1\\) and \\(1\\) with increasing frequency. Neither one-sided limit exists, and no value can be assigned to \\(f(0)\\) that would restore any form of continuity.
## Selected references
- **Harvard University N. Masson**. [Discontinuities and Monotonic Functions](https://people.math.harvard.edu/~nate/teaching/UPenn/2009/spring/math_360/lectures/week_8/lecture_14/lecture_14.pdf)
- **Harvard University O. Knill**. [Introduction to Functions and Calculus – Continuity](https://people.math.harvard.edu/~knill/teaching/math1a_2012/handouts/14-continuity.pdf)
- **UC Berkeley A. Vizeff**. [Lecture 5: Continuity and Discontinuities](https://math.berkeley.edu/~avizeff/calculus-I-F22/lecture-5.pdf)
