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Set of Continuous Functions

Source: Romanian District Olympiad 2018 - Grade XII - Problem 1

March 10, 2018
functioncontinuitysupremum

Problem Statement

Let F\mathcal{F} be the set of continuous functions f:[0,1]→Rf : [0, 1]\to\mathbb{R} satisfying max⁡0≤x≤1∣f(x)∣=1\max_{0\le x\le 1} |f(x)| = 1 and let I:F→RI : \mathcal{F} \to \mathbb{R},
I(f)=∫01f(x) dx−f(0)+f(1).I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).
a) Show that I(f)<3I(f) < 3, for any f∈Ff \in \mathcal{F}.
b) Determine sup⁡{I(f)∣f∈F}\sup\{I(f) \mid f \in \mathcal{F}\}.
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