Intermediate Value Theorem

Theorem
Let $$I$$ be a real interval.

Let $$a, b \in I$$ such that $$\left({a \, . \, . \, b}\right)$$ is an open interval.

Let $$f: I \to \mathbb{R}$$ be a real function which is continuous on $$\left({a \, . \, . \, b}\right)$$.

Let $$k \in \mathbb {R}$$ lie between $$f \left({a}\right)$$ and $$f \left({b}\right)$$.

That is, either:
 * $$f \left({a}\right) < k < f \left({b}\right)$$;
 * $$f \left({b}\right) < k < f \left({a}\right)$$.

Then $$\exists c \in \left({a \, . \, . \, b}\right)$$ such that $$f \left({c}\right) = k$$.

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

From Image of Interval by Continuous Function, the image of $$\left({a \, . \, . \, b}\right)$$ under $$f$$ is also an real interval.

Thus if $$k$$ lies between $$f \left({a}\right)$$ and $$f \left({b}\right)$$, it must be the case that $$k \in \mathrm{Im} \left({\left({a \,. \, . \, b}\right)}\right)$$.

The result follows.