MATHEMATICAL ANALYSIS 05
Intermediate Value Theorem
A continuous function cannot move from one height to another without passing through every height between them.
The theorem
If \(f\) is continuous on \([a,b]\) and \(K\) lies between \(f(a)\) and \(f(b)\), then there exists some \(c\in[a,b]\) such that
\[\boxed{f(c)=K}\]
Root existence
A particularly useful case is \(K=0\). If \(f(a)\lt0\lt f(b)\), continuity guarantees at least one \(c\) with \(f(c)=0\).
What it does not say
The theorem guarantees existence, not uniqueness and not the exact location of \(c\).
Why continuity is essential
A discontinuous function can jump from below \(K\) to above \(K\) without ever taking the value \(K\).