Suppose a continuous function satisfies

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

Then $f$ changes sign on $[a,b]$, so the intermediate value theorem guarantees at least one root in the interval.

The bisection method repeatedly evaluates the midpoint

$$m=\frac{a+b}{2}$$

and keeps the half-interval that still contains a sign change.

After $n$ iterations, the interval width is

$$\frac{b-a}{2^n},$$

so the root is bracketed with a directly controlled error.

Bisection is slow compared with some derivative-based methods, but its convergence is robust when a valid sign-changing bracket is maintained. Its strength is the guarantee: each iteration preserves a mathematical certificate that a root remains inside the current interval.