Derivative of differentiable function satisfies intermediate value property

Statement

Suppose $f$ is a differentiable function on an open interval $(p,q)$. Then, the derivative $f^{\prime }$ satisfies the intermediate value property on $(p,q)$: for $a<b$, both in $(p,q)$, and any value $w$ is strictly between $f^{\prime }(a)$ and $f^{\prime }(b)$, there exists $c\in (a,b)$ such that $f^{\prime }(c)=w$.

Related facts

• Intermediate value theorem: This states that any continuous function satisfies the intermediate value property.

Facts used

1. Lagrange mean value theorem

Proof

Proof idea

The proof idea is to find a difference quotient that takes the desired value intermediate between $f^{\prime }(a)$ and $f^{\prime }(b)$, then use Fact (1).

Proof details

Given: $f$ is a differentiable function on an open interval $(p,q)$. Suppose $a<b$, both in $(p,q)$, and suppose $w$ is strictly between $f^{\prime }(a)$ and $f^{\prime }(b)$,

To prove: There exists $c\in (a,b)$ such that $f^{\prime }(c)=w$.

Proof: Consider the function $g$ defined on the interval $[0,1]$ as follows:

$g(t):=\{\begin{array}{cc}f\text{'}(a),& t=0\\ ,& 0<t\le 1/2\\ ,& 1/2<t<1\\ f\text{'}(b),& t=1\\ \end{array}$