[!]. This exercise is used in Section e14.2. Suppose that [a, b] is a closed, nondegenerate interval and f : [a, b] ---7 R is bounded.

(a) Prove that if f is continuous at Xo E [a, b] and f(xo) =1= 0, then

(L) lb If(x)1 dx > 0.

(b) Show that if f is continuous on [a, b], then J: If(x)1 dx = ° if and only if f(x) = ° for all x E [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.