A famous result in classical analysis is the intermediate value theorem: Suppose $f$ is a continuous function on the interval $[a,b]$ such that $f(a)0$ and $f(b)>0$. Then there exists $x \in (a,b)$ such that $f(x)=0$. This theorem isn't valid in constructivist mathematics, but the following slightly weaker version is: Suppose $f$ is a continuous function on the interval $[a,b]$ such that $f(a)0$ and $f(b)>0$. Then for each $\epsilon >0$ there exists $x \in (a,b)$ such that $|f(x)|\epsilon$. This result uses the same conditions as the intermediate value theorem, but has a weaker conclusion. It is also possible to use slightly stronger conditions and then come up with a constructivist result that has the same conclusions as the intermediate value theorem: Suppose $f$ is a continuous function on the interval $[a,b]$ such that $f(a)0$ and $f(b)>0$. Also suppose that $f$ is locally non-zero: for every $x\in[a,b]$ and $\epsilon>0$ there exists $y$ such that $|x-y| \epsilon$ and $f(y)\neq 0$. Then there exists $x \in (a,b)$ such that $f(x)=0$.

Thus constructivist analysis doesn't have a result that is quite as strong as the classical intermediate value theorem, but it does have results that come close to it and are useful in many of the situations in which you might want to use the intermediate value theorem.

Back to main article