(Solution) - Suppose that a b and that f a -(2025 Original AI-Free Solution)

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Suppose that a < b and that f : [a, b] ? R is bounded.
a) Prove that if f is continuous at x0 ? [a, b] and f(x0) ? 0, then

b) Show that if f is continuous on [a, b], then ?ba |f(x)dx = 0 if and only if f(x) = 0 for all x ? [a, b].
c) Does part b) hold if the absolute values are removed? If it does, prove it. If it does not, provide a counterexample.