Intermediate Value Theorem

Theorem
Let $f: S \to \R$ be a real function on some subset $S$ of $\R$.

Let $I \subseteq S$ be a real interval.

Let $f: I \to \R$ be continuous on $I$.

Then $f$ is a Darboux function.

That is:

Let $a, b \in I$.

Let $k \in \R$ lie between $\map f a$ and $\map f b$.

That is, either:
 * $\map f a < k < \map f b$

or:
 * $\map f b < k < \map f a$

Then $\exists c \in \openint a b$ such that $\map f c = k$.

Proof
This theorem is a restatement of Image of Interval by Continuous Function is Interval.

From Image of Interval by Continuous Function is Interval, the image of $\openint a b$ under $f$ is also a real interval (but not necessarily open).

Thus if $k$ lies between $\map f a$ and $\map f b$, it must be the case that:
 * $k \in \Img {\openint a b}$

The result follows.

Also known as
This result is also known as Bolzano's theorem, for.

Some sources attribute it to, and call it the Weierstrass Intermediate Value Theorem.

Also see

 * Intermediate Value Theorem (Topology), of which this is a corollary