Intermediate Values and Root Bracketing — Cheat sheet

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Core rule

For continuous ff on [a,b][a,b], every value between the endpoint outputs is attained. Opposite signs guarantee an interior root.

Watch for

No uniqueness follows without another argument. A discontinuity invalidates the guarantee. Bisection midpoint error after nn halvings is at most (b−a)/2n+1(b-a)/2^{n+1}.