page_title: Inverse function theorem
section_title: Counter-example
hierarchical_section_title: Inverse function theorem / Counter-example
caption_attribution_description: English: Counter-example to the Inverse Function Theorem without the assumption of continuous differentiability
context_page_description: In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function.
  In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.
context_section_description: If one drops the assumption that the derivative is continuous, the function no longer need be invertible. For example  and  has discontinuous derivative
   and , which vanishes arbitrarily close to . These critical points are local max/min points of , so  is not one-to-one (and not invertible) on any interval containing . Intuitively, the slope  does not propagate to nearby points, where the slopes are governed by a weak but rapid oscillation.
caption: The function 
    
      
        
          f
          (
          x
          )
          =
          x
          +
          2
          
            x
            
              2
            
          
          sin
          ⁡
          (
          
            
              
                1
                x
              
            
          
          )
        
      
      {\displaystyle f(x)=x+2x^{2}\sin({\tfrac {1}{x}})}
    
   is bounded inside a quadratic envelope near the line 
    
      
        
          y
          =
          x
        
      
      {\displaystyle y=x}
    
  , so 
    
      
        
          
            f
            ′
          
          (
          0
          )
          =
          1
        
      
      {\displaystyle f'(0)=1}
    
  . Nevertheless, it has local max/min points accumulating at 
    
      
        
          x
          =
          0
        
      
      {\displaystyle x=0}
    
  , so it is not one-to-one on any surrounding interval.
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