Darboux's theorem (analysis)

In real analysis, Darboux's theorem states that the derivative of any real-valued function of a real variable has the intermediate value property, that is, that the image of an interval is also an interval.

When

f

{\displaystyle f}

is continuously differentiable, this is a consequence of the intermediate value theorem. But even when

f

′

{\displaystyle f'}

is not continuous, Darboux's theorem places a restriction on the behaviour of

f

′

{\displaystyle f'}

over any closed interval.

Statement of the theorem

Let

I

{\displaystyle I}

be an open interval, and let

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}

.

The original proof by Jean Gaston Darboux was published in 1875.

Proofs

Proof from the extreme value theorem

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}

.