3.6.1—Root location

For a continuous f, opposite signs at a and b guarantee at least one root between them. A root-location interval is evidence of existence, not automatically uniqueness.

Evaluate f at the endpoints accurately, keep the interval ordered, and report its width or decimal bounds as the requested accuracy.

If f(1)=−0.4 and f(2)=0.3, continuity gives a root in (1,2); bisection can reduce this to a bracket of width below 0.01.

A sign change can contain several roots, and a root exactly at an endpoint should be stated separately.