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

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