Question 9
Let $\mathcal{F}$ be the set of all continuous functions $f: [0, 1] \to \mathbb{R}$, satisfying
$$ \max_{0 \le x \le 1} |f(x)| = 1, \text{ and let } I: \mathcal{F} \to \mathbb{R}, $$
$$
I(f) = \int_{0}^{1} f(x) \, dx - f(0) + f(1).
$$
$$ \max_{0 \le x \le 1} |f(x)| = 1, \text{ and let } I: \mathcal{F} \to \mathbb{R}, $$
$$
I(f) = \int_{0}^{1} f(x) \, dx - f(0) + f(1).
$$