(c.f. June 2012 #1b) If \(f:[0,1]\to[0,1]\) is continuous, show that \(f(x)=x\) for some \(x\in[0,1]\).
Solution
1. Consider \(g(x) = f(x) - x\) on \([0,1]\).Proof: \(g\) is continuous (difference of continuous functions).At the endpoints: \[g(0) = f(0) - 0 = f(0) \ge 0 \quad (\text{as } f([0,1]) \subseteq [0,1]),\] \[g(1) = f(1) - 1 \le 0 \quad (\text{as } f(1) \le 1).\]
2. Apply the intermediate value theorem.Proof: if \(g(0) = 0\) or \(g(1) = 0\), then \(0\) or \(1\) is a fixed point.Otherwise \(g(0) > 0\) and \(g(1) < 0\), so by IVT there is \(x \in (0,1)\) with \(g(x) = 0\), i.e. \(f(x) = x\).
3. Q.E.D.