User:Torarin/Sandbox

=Intermediate Value Theorem for Derivatives=

Theorem
Let $I$ be an open interval.

Let $f : I \to \R$ be everywhere differentiable.

Then $f'$ satisfies the Intermediate Value Property.

Proof
Since $\forall \paren {a, b \in I: a < b}: \openint a b \subseteq I$, the result follows from Image of Interval by Derivative.

=Image of Interval by Derivative=

Theorem
Let $f$ be a real function that is everywhere differentiable.

Let $I \subseteq \Dom f$ be an interval.

Then $f' \sqbrk I$ is an interval.

Proof
Let $x_1, x_2 \in f' \sqbrk I : x_1 < x_2$.

Let $\xi \in \closedint {x_1} {x_2}$.

We need to show that $\xi \in f' \sqbrk I$.

Let $a, b \in I : \map {f'} a = x_1 \land \map {f'} b = x_2$.

Assume $a < b$. The case $b < a$ is handled similarly.

Let $\map g x = \map f x - \xi x$.

Then $\map {g'} x = \map {f'} x - \xi$.

By Differentiable Function is Continuous, $g$ is continuous.

By Restriction of Continuous Mapping is Continuous, $g \restriction_{\closedint a b}$ is continuous.

From Corollary 3 to Continuous Image of Compact Space is Compact, $g\restriction_{\closedint a b}$ attains a minimum.

By hypothesis, $\map {g'} a < 0$ and $\map {g'} b > 0$.

Let $\map {N^*_\epsilon} x$ denote the deleted $\epsilon$-neighborhood of $x$.

Then from Behaviour of Function Near Limit:
 * $\exists \map {N^*_\epsilon} a : \forall x \in \map {N^*_\epsilon} a : \dfrac {\map g x - \map g a} {x - a} < 0$

and
 * $\exists \map {N^*_\delta} b : \forall x \in \map {N^*_\delta} b : \dfrac {\map g x - \map g b} {x - b} > 0$

Thus:
 * $\exists x \in \map {N^*_\epsilon} a \cap \closedint a b: \map g x < \map g a$

and
 * $\exists x \in \map {N^*_\delta} b \cap \closedint a b: \map g x < \map g b$

Hence $\map g a$ and $\map g b$ are not minima of $g \restriction_{\closedint a b}$, and so $g \restriction_{\closedint a b}$ must attain its minimum at some $m \in \openint a b$.

By Derivative at Maximum or Minimum, $\map {g'} m = 0$.

Hence $\map {f'} m = \xi$.

The result follows.