In essence, the Intermediate Value theorem states that a continuous function $f$ takes on every value between $f\left(a\right)$ and $f\left(b\right)$. This property of a continuous function is called the "Darboux" property. Hence, every continuous function has the Darboux property, but there are functions that have the Darboux property that are not continuous. As a counterexample, take the function whose rule is $\sin \left(1\/x\right)$ for $x\ne 0$, and has the value 0 at $x\=0$. This function takes on every value between $-1$ and 1, yet is not continuous.

As intuitive as the Intermediate Value theorem (IVT) might seem, it is equivalent to the far deeper completeness property of the real numbers, which essentially states that there are no gaps in the real number line. If there were, then the IVT might not hold because a crucial number could be missing from the range of $f$. So, asserting that $f$ takes on every possible value between $f\left(a\right)$ and $f\left(b\right)$ is an assertion that there are no gaps in the real line between these two values, that is, the real numbers are "complete."

For more than a century after Newton and Leibniz articulated the tools of the calculus, these tools were applied to a host of problems in physics and engineering, with spectacular success. It wasn't until much later that mathematicians began to consider such foundational issues as the IVT and its relation to completeness of the real numbers. However, theorems like the IVT are behind interesting observations even today. Click here for an article discussing the use of the IVT in proving that a "wobbly table" can be stabilized by a rotation about its center.