Working with the Intermediate Value Theorem (IVT) AB & BC

The Intermediate Value Theorem says that if $f$ is continuous on a closed interval $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then $f(c) = N$ for at least one $c$ in $(a,b)$. It is the formal version of the observation that an unbroken curve cannot get from one height to another without passing through every height between. Its standard use is showing an equation has a solution on an interval, where $N = 0$ is one case of a general statement.

Errors come in two shapes. The hypotheses go unchecked, most often by not verifying continuity on the closed interval, which fails outright when an asymptote sits inside, so the theorem gets applied to a function like $\frac{1}{x}$ on $[-1,1]$ and appears to prove a root that does not exist. Or the conclusion gets overread: it promises at least one input, so claims of a unique solution, of a particular location, or of a maximum go beyond what was established. Failing the hypotheses means no guarantee, not no root.