Nowhere continuous function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a nowhere continuous function, also called an everywhere discontinuous function, is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that is not <a href="/facts/Continuous_function/sKbl02pB">continuous</a> at any point of its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>. If 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a function from <a href="/facts/Real_number/R02gw5Pb">real numbers</a> to real numbers, then 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is nowhere continuous if for each point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 there is some 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
 such that for every 
  
    
      
        δ
        >
        0
        ,
      
    
    {\displaystyle \delta >0,}
  
 we can find a point 
  
    
      
        y
      
    
    {\displaystyle y}
  
 such that 
  
    
      
        
          |
        
        x
        −
        y
        
          |
        
        <
        δ
      
    
    {\displaystyle |x-y|<\delta }
  
 and 
  
    
      
        
          |
        
        f
        (
        x
        )
        −
        f
        (
        y
        )
        
          |
        
        ≥
        ε
      
    
    {\displaystyle |f(x)-f(y)|\geq \varepsilon }
  
. Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values.
</p><p>More general definitions of this kind of function can be obtained, by replacing the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> by the distance function in a <a href="/facts/Metric_space/gnBBRXtc">metric space</a>, or by using the definition of continuity in a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>.
</p>

Nowhere continuous function open-in-new

Nowhere continuous function