In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval.
 When ƒ is continuously differentiable (ƒ in C1([a,b])), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.
 Let 
  
    
      
        I
      
    
    {\displaystyle I}
  
 be a closed interval, 
  
    
      
        f
        :
        I
        →
        
          R
        
      
    
    {\displaystyle f\colon I\to \mathbb {R} }
  
 be a real-valued differentiable function. Then 
  
    
      
        
          f
          ′
        
      
    
    {\displaystyle f'}
  
 has the intermediate value property: If 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 are points in 
  
    
      
        I
      
    
    {\displaystyle I}
  
 with 
  
    
      
        a
        <
        b
      
    
    {\displaystyle a<b}
  
, then for every 
  
    
      
        y
      
    
    {\displaystyle y}
  
 between 
  
    
      
        
          f
          ′
        
        (
        a
        )
      
    
    {\displaystyle f'(a)}
  
 and 
  
    
      
        
          f
          ′
        
        (
        b
        )
      
    
    {\displaystyle f'(b)}
  
, there exists an 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 such that 
  
    
      
        
          f
          ′
        
        (
        x
        )
        =
        y
      
    
    {\displaystyle f'(x)=y}
  
.
 Proof 1. The first proof is based on the extreme value theorem.
 If 
  
    
      
        y
      
    
    {\displaystyle y}
  
 equals 
  
    
      
        
          f
          ′
        
        (
        a
        )
      
    
    {\displaystyle f'(a)}
  
 or 
  
    
      
        
          f
          ′
        
        (
        b
        )
      
    
    {\displaystyle f'(b)}
  
, then setting 
  
    
      
        x
      
    
    {\displaystyle x}
  
 equal to 
  
    
      
        a
      
    
    {\displaystyle a}
  
 or 
  
    
      
        b
      
    
    {\displaystyle b}
  
, respectively, gives the desired result. Now assume that 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is strictly between 
  
    
      
        
          f
          ′
        
        (
        a
        )
      
    
    {\displaystyle f'(a)}
  
 and 
  
    
      
        
          f
          ′
        
        (
        b
        )
      
    
    {\displaystyle f'(b)}
  
, and in particular that 
  
    
      
        
          f
          ′
        
        (
        a
        )
        >
        y
        >
        
          f
          ′
        
        (
        b
        )
      
    
    {\displaystyle f'(a)>y>f'(b)}
  
. Let  
  
    
      
        φ
        :
        I
        →
        
          R
        
      
    
    {\displaystyle \varphi \colon I\to \mathbb {R} }
  
 such that 
  
    
      
        φ
        (
        t
        )
        =
        f
        (
        t
        )
        −
        y
        t
      
    
    {\displaystyle \varphi (t)=f(t)-yt}
  
. If it is the case that 
  
    
      
        
          f
          ′
        
        (
        a
        )
        <
        y
        <
        
          f
          ′
        
        (
        b
        )
      
    
    {\displaystyle f'(a)<y<f'(b)}
  
 we adjust our below proof, instead asserting that 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 has its minimum on 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
.
 Since 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 is continuous on the closed interval 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
, the maximum value of 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 on 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 is attained at some point in 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
, according to the extreme value theorem.
 Because 
  
    
      
        
          φ
          ′
        
        (
        a
        )
        =
        
          f
          ′
        
        (
        a
        )
        −
        y
        >
        0
      
    
    {\displaystyle \varphi '(a)=f'(a)-y>0}
  
, we know 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 cannot attain its maximum value at 
  
    
      
        a
      
    
    {\displaystyle a}
  
. (If it did, then 
  
    
      
        (
        φ
        (
        t
        )
        −
        φ
        (
        a
        )
        )
        
          /
        
        (
        t
        −
        a
        )
        ≤
        0
      
    
    {\displaystyle (\varphi (t)-\varphi (a))/(t-a)\leq 0}
  
 for all 
  
    
      
        t
        ∈
        (
        a
        ,
        b
        ]
      
    
    {\displaystyle t\in (a,b]}
  
, which implies 
  
    
      
        
          φ
          ′
        
        (
        a
        )
        ≤
        0
      
    
    {\displaystyle \varphi '(a)\leq 0}
  
.)
 Likewise, because 
  
    
      
        
          φ
          ′
        
        (
        b
        )
        =
        
          f
          ′
        
        (
        b
        )
        −
        y
        <
        0
      
    
    {\displaystyle \varphi '(b)=f'(b)-y<0}
  
, we know 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 cannot attain its maximum value at 
  
    
      
        b
      
    
    {\displaystyle b}
  
.
 Therefore, 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 must attain its maximum value at some point 
  
    
      
        x
        ∈
        (
        a
        ,
        b
        )
      
    
    {\displaystyle x\in (a,b)}
  
. Hence, by Fermat's theorem, 
  
    
      
        
          φ
          ′
        
        (
        x
        )
        =
        0
      
    
    {\displaystyle \varphi '(x)=0}
  
, i.e. 
  
    
      
        
          f
          ′
        
        (
        x
        )
        =
        y
      
    
    {\displaystyle f'(x)=y}
  
.
 Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.
 Define 
  
    
      
        c
        =
        
          
            1
            2
          
        
        (
        a
        +
        b
        )
      
    
    {\displaystyle c={\frac {1}{2}}(a+b)}
  
.
For 
  
    
      
        a
        ≤
        t
        ≤
        c
        ,
      
    
    {\displaystyle a\leq t\leq c,}
  
 define 
  
    
      
        α
        (
        t
        )
        =
        a
      
    
    {\displaystyle \alpha (t)=a}
  
 and 
  
    
      
        β
        (
        t
        )
        =
        2
        t
        −
        a
      
    
    {\displaystyle \beta (t)=2t-a}
  
.
And for 
  
    
      
        c
        ≤
        t
        ≤
        b
        ,
      
    
    {\displaystyle c\leq t\leq b,}
  
 define 
  
    
      
        α
        (
        t
        )
        =
        2
        t
        −
        b
      
    
    {\displaystyle \alpha (t)=2t-b}
  
 and 
  
    
      
        β
        (
        t
        )
        =
        b
      
    
    {\displaystyle \beta (t)=b}
  
.
 Thus, for 
  
    
      
        t
        ∈
        (
        a
        ,
        b
        )
      
    
    {\displaystyle t\in (a,b)}
  
 we have 
  
    
      
        a
        ≤
        α
        (
        t
        )
        <
        β
        (
        t
        )
        ≤
        b
      
    
    {\displaystyle a\leq \alpha (t)<\beta (t)\leq b}
  
.
Now, define  
  
    
      
        g
        (
        t
        )
        =
        
          
            
              (
              f
              ∘
              β
              )
              (
              t
              )
              −
              (
              f
              ∘
              α
              )
              (
              t
              )
            
            
              β
              (
              t
              )
              −
              α
              (
              t
              )
            
          
        
      
    
    {\displaystyle g(t)={\frac {(f\circ \beta )(t)-(f\circ \alpha )(t)}{\beta (t)-\alpha (t)}}}
  
 with 
  
    
      
        a
        <
        t
        <
        b
      
    
    {\displaystyle a<t<b}
  
.

  
    
      
        
        g
      
    
    {\displaystyle \,g}
  
 is continuous in 
  
    
      
        (
        a
        ,
        b
        )
      
    
    {\displaystyle (a,b)}
  
.
 Furthermore, 
  
    
      
        g
        (
        t
        )
        →
        
          
            f
          
          ′
        
        (
        a
        )
      
    
    {\displaystyle g(t)\rightarrow {f}'(a)}
  
 when 
  
    
      
        t
        →
        a
      
    
    {\displaystyle t\rightarrow a}
  
 and 
  
    
      
        g
        (
        t
        )
        →
        
          
            f
          
          ′
        
        (
        b
        )
      
    
    {\displaystyle g(t)\rightarrow {f}'(b)}
  
 when 
  
    
      
        t
        →
        b
      
    
    {\displaystyle t\rightarrow b}
  
; therefore, from the Intermediate Value Theorem, if 
  
    
      
        y
        ∈
        (
        
          
            f
          
          ′
        
        (
        a
        )
        ,
        
          
            f
          
          ′
        
        (
        b
        )
        )
      
    
    {\displaystyle y\in ({f}'(a),{f}'(b))}
  
 then, there exists 
  
    
      
        
          t
          
            0
          
        
        ∈
        (
        a
        ,
        b
        )
      
    
    {\displaystyle t_{0}\in (a,b)}
  
 such that 
  
    
      
        g
        (
        
          t
          
            0
          
        
        )
        =
        y
      
    
    {\displaystyle g(t_{0})=y}
  
.
Let's fix 
  
    
      
        
          t
          
            0
          
        
      
    
    {\displaystyle t_{0}}
  
.
 From the Mean Value Theorem, there exists a point 
  
    
      
        x
        ∈
        (
        α
        (
        
          t
          
            0
          
        
        )
        ,
        β
        (
        
          t
          
            0
          
        
        )
        )
      
    
    {\displaystyle x\in (\alpha (t_{0}),\beta (t_{0}))}
  
 such that 
  
    
      
        
          
            f
          
          ′
        
        (
        x
        )
        =
        g
        (
        
          t
          
            0
          
        
        )
      
    
    {\displaystyle {f}'(x)=g(t_{0})}
  
.
Hence, 
  
    
      
        
          
            f
          
          ′
        
        (
        x
        )
        =
        y
      
    
    {\displaystyle {f}'(x)=y}
  
.
 A Darboux function is a real-valued function ƒ which has the "intermediate value property": for any two values a and b in the domain of ƒ, and any y between ƒ(a) and ƒ(b), there is some c between a and b with ƒ(c) = y.  By the intermediate value theorem, every continuous function on a real interval is a Darboux function. Darboux's contribution was to show that there are discontinuous Darboux functions.
 Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the left hand and right hand limits does not exist.
 An example of a Darboux function that is discontinuous at one point is the topologist's sine curve function:
 By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function 
  
    
      
        x
        ↦
        
          x
          
            2
          
        
        sin
        ⁡
        (
        1
        
          /
        
        x
        )
      
    
    {\displaystyle x\mapsto x^{2}\sin(1/x)}
  
 is a Darboux function even though it is not continuous at one point.
 An example of a Darboux function that is nowhere continuous is the Conway base 13 function.
 Darboux functions are a quite general class of functions. It turns out that any real-valued function ƒ on the real line can be written as the sum of two Darboux functions. This implies in particular that the class of Darboux functions is not closed under addition.
 A strongly Darboux function is one for which the image of every (non-empty) open interval is the whole real line.  The Conway base 13 function is again an example.