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Suppose random continuous real functions have unique zeros. If pointwise convergence in probability gives opposite strict limiting signs at the two ends of every sufficiently small interval about the target, the intermediate value theorem traps their zeros in those intervals with probability tending to one. The limiting function need not be continuous and the random functions need not be monotone.

If $E\subseteq [0,1]$ is a Lebesgue measurable set and $0\le b\le \lambda (E)$, some measurable subset $B\subseteq E$ has Lebesgue measure $b$. The function $F(t)=\lambda (E\cap [0,t])$ has Lipschitz continuity with constant $1$, starts at $0$ and ends at $\lambda (E)$; the intermediate value theorem proves the assertion. Applying the assertion successively to remaining portions of $E$ gives pairwise disjoint measurable pieces with any finite list of nonnegative measures whose sum is at most $\lambda (E)$. This is a concrete consequence of non-atomic measure structure, which fails for general measures containing an atom of a measure.

For continuous $f:[0,1]\to [0,1]$, the function $f(x)-x$ is nonnegative at zero and nonpositive at one, so the intermediate value theorem gives a fixed point.

If $f:[0,1]\to \mathbb{R}$ is continuous and $f(0)=f(1)$, then for every positive integer $n$ there is $\alpha \in [0,1-1/n]$ with $f(\alpha +1/n)=f(\alpha )$. For $n>1$, the continuous difference $h_n(x)=f(x+1/n)-f(x)$ satisfies ${\sum }_{j=0}^{n-1}{h}_n(j/n)=0$. Either one sampled difference is zero or two have opposite signs; the intermediate value theorem then supplies a zero between them. For $n=1$, the endpoint chord suffices.

For closed intervals $J$ and $K$, write $J⟶K$ under $F$ when $F(J)\supseteq K$. The intermediate value theorem implies that there is a closed subinterval $L\subseteq J$ with $F(L)=K$.

A continuous injective real-valued function on a real interval is strictly monotone. A value outside the interval between two endpoint values would, by the intermediate value theorem, create a repeated value. Applying this observation to subintervals gives consistent strict ordering. If the left endpoint value is smaller than the right endpoint value, the function is strictly increasing; otherwise it is strictly decreasing.

The Möbius line bundle over the circle iswith projection to $t$ modulo $2\pi$. It is a smooth rank-one vector bundle. A section is represented by a function $s$ satisfying $s(t+2\pi )=-s(t)$, so the intermediate value theorem forces every continuous section to vanish somewhere. Thus it has no nowhere-zero section and is not isomorphic to the trivial vector bundle of rank one.

A symmetric triangle with direct calls and two-link overflow routes has Erlang fixed point approximation equation $B=E(\lambda [1+2B(1-B)],C)$. For $C=1000$ and $\lambda =950$, this has at least three solutions, certified by the intermediate value theorem using signs at $B=0,0.01,0.1,0.4$.