Real induction (statement)

Principle (real induction). Let $S\subseteq[a,b]$ satisfy:
1. $a\in S$;
2. If $x\in S$ and $x<b$, then $[x,x+\varepsilon)\subseteq S$ for some $\varepsilon>0$;
3. If $[a,x)\subseteq S$ with $x\le b$, then $x\in S$.
Then $S=[a,b]$.

Application. Proving that a continuous function has the intermediate value property.

Conclusion. Real induction is a powerful tool for proofs on intervals.