Image of Interval by Derivative

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

Let $I \subseteq \operatorname{Dom} \left({f}\right)$ be an interval.

Then $f' \left({I}\right)$ is an interval.

Proof
Let $x_1, x_2 \in f' \left({I}\right): x_1 < x_2$.

Let $\xi \in \left({x_1 \,.\,.\, x_2}\right)$.

We need to show that $\xi \in f' \left({I}\right)$.

Let $a, b \in I : f' \left({a}\right) = x_1 \land f' \left({b}\right) = x_2$.

, assume $a < b$. The case $b < a$ is handled similarly.

Let $g \left({x}\right) = f \left({x}\right) - \xi x$.

Then:
 * $g' \left({x}\right) = f' \left({x}\right) - \xi$

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

By Restriction of Continuous Mapping is Continuous, $g \restriction_{\left[{a \,.\,.\, b}\right]}$ is continuous.

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

By hypothesis, $g' \left({a}\right) < 0$ and $g' \left({b}\right) > 0$.

Let $N^*_\epsilon \left({x}\right)$ denote the deleted $\epsilon$-neighborhood of $x$.

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

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

Thus:
 * $\exists x \in N^*_\epsilon \left({a}\right) \cap \left[{a \,.\,.\, b}\right]: g \left({x}\right) < g \left({a}\right)$

and
 * $\exists x \in N^*_\delta \left({b}\right) \cap \left[{a \,.\,.\, b}\right]: g \left({x}\right) < g \left({b}\right)$

Hence $g \left({a}\right)$ and $g \left({b}\right)$ are not minima of $g\restriction_{\left[{a \,.\,.\, b}\right]}$.

So $g \restriction_{\left[{a \,.\,.\, b}\right]}$ must attain its minimum at some $m \in \left({a \,.\,.\, b}\right)$.

By Derivative at Maximum or Minimum, $g' \left({m}\right) = 0$.

Hence $f' \left({m}\right) = \xi$.

The result follows.