= Interval root exclusion
{title2=$0\notin F(X)\Rightarrow\forall x\in X:\ f(x)\ne0$}
<Interval root exclusion> certifies that a <function> has no <zero of a function> on $X$ when a valid <interval enclosure> $F(X)$ excludes zero. For $g(x)=x(1-x)-1/3$ on $[0,1]$, the <interval square range> yields $G(X)=[-1/3,-1/12]$, which excludes zero. Direct <interval arithmetic> yields $[-1/3,2/3]$ and is inconclusive, even though the <function> has no <zero of a function> there.
Knowing only that a calculated <interval enclosure> contains zero does not certify a <zero of a function>: it may contain unattainable values. Even the positive <constant function> $f(x)=1$ has a valid, loose <interval enclosure> $[-1,2]$. A separate existence argument, such as reliable opposite endpoint signs together with <continuity> in the <intermediate value theorem>, is required; see <root bracket> and <bisection method>. <Interval subdivision> can resolve some inconclusive pieces, but remaining pieces must be reported as unresolved unless an existence argument succeeds. This makes <interval root exclusion> useful for systematically discarding regions in a search.
// Target: analysis.bigb
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