Let g be continuous on [0,1] with 0≤ g(x)≤1. To prove that g(c)=c for some c, to which function is the theorem applied?

Exercise · ZW-INTERMEDIATE-VALUE-THEOREM-016 · ★★☆ Intermediate

Let \(g\) be continuous on \([0,1]\) with \(0\le g(x)\le1\). To prove that \(g(c)=c\) for some \(c\), to which function is the theorem applied?

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\(h(0)=g(0)\ge0\) and \(h(1)=g(1)-1\le0\); the theorem gives \(c\) with \(h(c)=0\), i.e. \(g(c)=c\). The other functions do not encode that equality.

If \(f\) is continuous on \([a,b]\) and \(N\) lies between \(f(a)\) and \(f(b)\), there is some \(c\in(a,b)\) with \(f(c)=N\).

\[f\ \text{continuous on }[a,b],\ f(a)