kb/data/en.wikipedia.org/wiki/Classification_of_discontinuities-3.md

---
title: "Classification of discontinuities"
chunk: 4/4
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"
---

Therefore if 
  
    
      
        
          x
          
            0
          
        
        ∈
        I
      
    
    {\displaystyle x_{0}\in I}
  
 is a discontinuity of a derivative function 
  
    
      
        f
        :
        I
        →
        
          R
        
      
    
    {\displaystyle f:I\to \mathbb {R} }
  
, then necessarily 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
 is a fundamental essential discontinuity of 
  
    
      
        f
      
    
    {\displaystyle f}
  
.
Notice also that when  
  
    
      
        I
        =
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle I=[a,b]}
  
 and 
  
    
      
        f
        :
        I
        →
        
          R
        
      
    
    {\displaystyle f:I\to \mathbb {R} }
  
 is a bounded function, as in the assumptions of Lebesgue's theorem, we have for all 
  
    
      
        
          x
          
            0
          
        
        ∈
        (
        a
        ,
        b
        )
      
    
    {\displaystyle x_{0}\in (a,b)}
  
:

  
    
      
        
          lim
          
            x
            →
            
              x
              
                0
              
              
                ±
              
            
          
        
        f
        (
        x
        )
        ≠
        ±
        ∞
        ,
      
    
    {\displaystyle \lim _{x\to x_{0}^{\pm }}f(x)\neq \pm \infty ,}
  

  
    
      
        
          lim
          
            x
            →
            
              a
              
                +
              
            
          
        
        f
        (
        x
        )
        ≠
        ±
        ∞
        ,
      
    
    {\displaystyle \lim _{x\to a^{+}}f(x)\neq \pm \infty ,}
  
 and

  
    
      
        
          lim
          
            x
            →
            
              b
              
                −
              
            
          
        
        f
        (
        x
        )
        ≠
        ±
        ∞
        .
      
    
    {\displaystyle \lim _{x\to b^{-}}f(x)\neq \pm \infty .}
  

Therefore any essential discontinuity of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a fundamental one.

== See also ==
Removable singularity – Undefined point on a holomorphic function which can be made regular
Mathematical singularity – Point where a mathematical object behaves irregularlyPages displaying short descriptions of redirect targets
Extension by continuity – Property of topological space
Smoothness – Number of derivatives of a function (mathematics)
Geometric continuity – Number of derivatives of a function (mathematics)Pages displaying short descriptions of redirect targets
Parametric continuity – Number of derivatives of a function (mathematics)Pages displaying short descriptions of redirect targets

== Notes ==

== References ==

== Sources ==
Malik, S.C.; Arora, Savita (1992). Mathematical Analysis (2nd ed.). New York: Wiley. ISBN 0-470-21858-4.{{cite book}}:  CS1 maint: publisher location (link)

== External links ==
"Discontinuous". PlanetMath.
"Discontinuity" by Ed Pegg, Jr., The Wolfram Demonstrations Project, 2007.
Weisstein, Eric W. "Discontinuity". MathWorld.
Kudryavtsev, L.D. (2001) [1994]. "Discontinuity point". Encyclopedia of Mathematics. EMS Press.