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Let \(f \in C([0, 1], \mathbb{R})\) with \(0 \le f \le 1\). Show that \(0 \le \bar f \le 1\), and show there exists \(c \in [0, 1]\) with \(f(c) = \bar f\). Hint: use P3.3(ii) (monotonicity) for the bounds on \(\bar f\), then apply P3.4 (second clause: existence of \(c\) when \(f\) is continuous).
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