If a function $f$ is continuous on a closed interval $[a,b]$, then it takes every value between $f(a)$ and $f(b)$ somewhere in that interval.

In particular, if

$$f(a)<0<f(b)$$

or

$$f(b)<0<f(a),$$

then there is at least one point $c\in(a,b)$ such that

$$f(c)=0.$$

The theorem guarantees existence, not uniqueness and not a formula for the point $c$.

Continuity is essential: a discontinuous function can jump from a negative value to a positive value without ever taking the value zero.

The intermediate value theorem turns a sign change into a rigorous root-existence certificate and underlies bracketing methods such as bisection.