kb/data/en.wikipedia.org/wiki/Classification_of_discontinuities-0.md

---
title: "Classification of discontinuities"
chunk: 1/4
source: "https://en.wikipedia.org/wiki/Classification_of_discontinuities"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:08:17.602531+00:00"
instance: "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.