= Residual root error bound
{title2=$|\widehat x-x_*|\leq|f(\widehat x)|/m$}
Suppose $f$ is continuous on an <interval>, differentiable in its interior, and $|f'|\geq m>0$ throughout the interior. If $x_*$ is a <root> in the <interval> and $\widehat x$ is another point in it, the <mean value theorem> gives $f(\widehat x)-f(x_*)=f'(\xi)(\widehat x-x_*)$ for an intermediate point $\xi$. Taking absolute values proves the stated bound from the <residual of a root equation>. This converts an observed residual into a stopping certificate, provided the evaluation error in that residual is also controlled.
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