2. Show that the followings are not uniformly continuous on the given do- mains. (a) g(a) = r2 on R. (b) h(r) = for r> 0. %3D
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- Suppose f : [a,∞) → R is continuous and f(x) → 0 as x →∞. Prove that f is uniformly continuous on [a,∞).Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is Lipschitz continuous on (−∞, ∞).If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the following is FALSE?
- Suppose f is differentiable on (0,1) and that there exists M > 0 so that|f′(x)|≤M for all x ∈(0,1).Prove that f is uniformly continuous on (0,1).Suppose g : [a, b] → R is continuous except at x_0 ∈ (a, b) and bounded. Prove thatg ∈ R(x) on [a, b].Suppose that w and r are continuous functions on (−∞, ∞), W (x) is an invertible antiderivative of w(x), and R(x) is an antiderivative of r(x). Circle all of the statements that must be true.