Suppose that a < b and that f : [a, b] → R is continuous. Show thatfor all c ∈ [a, b] i...

Suppose that a < b and that f : [a, b] → R is continuous. Show that

for all c ∈ [a, b] if and only if f (x) = 0 for all x ∈ [a, b]. (Compare with Exercise, and notice that f need not be nonnegative here.)

Exercise

This exercise is used in Section. 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

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b) Show that if f is continuous on [a, b], then  if and only if f (x) = 0 for all x ∈ [a, b].

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c) Does part b) hold if the absolute values are removed? If it does, prove it. If it does not, provide a counterexample.