= Intermediate value theorem hypotheses and counterexamples
Three explicit <functions> separate the hypotheses from the conclusion. These are original mathematical illustrations, not measured data.
The <polynomial function> $h(x)=x^3-x=x(x-1)(x+1)$ is a <continuous function> on $[-3/2,3/2]$, with $h(-3/2)=-15/8$ and $h(3/2)=15/8$. The <intermediate value theorem> guarantees at least one <zero of a function> in the interior, while the <polynomial factorization> displays three: $-1,0,1$. Thus strict opposite signs do not establish uniqueness.
The everywhere-defined <function>
$$
j(x)=\begin{cases}-1,&-1\leq x<0,\\1,&0\leq x\leq1\end{cases}
$$
has $j(-1)=-1$ and $j(1)=1$, yet no <zero of a function>. Its <jump discontinuity> at zero is exactly where the <continuity> hypothesis fails. A drawn vertical connector would be misleading: the <function graph> contains neither the intermediate heights nor a <zero of a function> at that jump. Even <continuity> only on the interior is insufficient: setting $q(x)=-1$ for $0\leq x<1$ and $q(1)=1$ gives opposite endpoint signs on $[0,1]$ with no <zero of a function>, because the left <one-sided limit> at $1$ is not $q(1)$.
The <polynomial function> $t(x)=x^2$ on $[-1,1]$ is a <continuous function>, and $t(-1)=t(1)=1$, yet $t(0)=0$. Its <root of a polynomial> at zero has <multiplicity of a root> two. The <function graph> touches its horizontal <tangent line> there and stays positive on both sides: existence of a <zero of a function> does not require a sign change. More generally, near a real <root of a polynomial> $r$, a <polynomial factorization> $p(x)=(x-r)^{2m}u(x)$ with $m\geq1$ and $u(r)\neq0$ has the same sign on both sides sufficiently close to $r$, since the <continuous function> $u$ retains its sign. The <intermediate value theorem> does not rule out values outside the range between endpoint outputs.<Calculus Volume 1 by Gilbert Strang and others>[§2.4, Example 2.37: positive endpoint values do not exclude an interior zero; Theorem 2.11 requires continuity on the whole interval]{ref}
\Image[/intermediate-value-continuous-jump-tangent.png]
{title=Intermediate value hypotheses: three continuous roots, a jump with no root, and a double root without a sign change}
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