3. Suppose that r(t) < s(t) on [1, ), and r(t) is continuous and strictly decreasing on [1, 0). If Ss(t)dt is converge, then lim r(n) = 0. m

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Q: Exercise 3. Find f(x) = lim,0 fn(x) on S. Show that (fn)n converges uniformly to f on closed subsets…

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Q: (4) Directly from the definition of continuity, show that f: Ro→ R defined by f(x) = 1/2 is…

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Q: (Group B) Suppose f: [0, 1] → R is a continuous function such that f(0) = f(1). Prove [0,1] such…

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Q: Express a = the limit lim Σ(4(x*)² – 3(x* )³) Aï; over [6, 7] as an integral. n→∞ i=1 Provide a, b…

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Q: Let fn (x) uniformly converge to f(x) on [a, b]. If fn(x) E R[a, b], then f(x) E R[a, b] True False

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Q: Let f: R→ R be a function such that f(x) ≥ x, for all x E R. Show that if Ef (bn) converge, then Σb,…

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Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

Q: Let f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

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Q: True or false: If f: R\ {0} → R is uniformly continuous, then lim f(x) 2-0 exists. True False

A: f: ℝ\0→ℝ is a uniform continuous function , We have to verify whether limx→0fx exist or not

Q: 4. Let f be a function continuous on [0, 1] and twice differentiable on (0,1). (a) Suppose that = 0…

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Q: Let f be a continuous function on [0, 1] with f(0) = f(1). (a) By considering g(x) = f(x) f(x + ¹)…

A: Sol:- (a) Since f(0) = f(1), we have g(0) = f(0) - f12 = f(1) - f12 = -g12. Since g is continuous on…

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Q: 6. Let f: Q R be defined by f(x) Show that f is continuous. 0, :{ 1, = if x € (-∞, √2) nQ if x €…

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Q: Show that a continuous function f : [0, 1] → Z is a constant function.

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Q: 6. Let f(x) = x2/3. (a) Then fis uniformly continuous on [-1,1]. Why? (b) Show that there does not…

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Q: 11. (a) Let F(s) = L{f(t)}, where f(t) is piecewise continuous and of exponential order on [0, ∞o).…

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Q: If ƒ is continuous for a < x < b and differentiable for a < x < b , which of the following could be…

A: Here, given that f is continuous for a≤x≤b and differentiable for a&lt;x&lt;b.

Q: Define for n ≥ 1, fn(x) = limn→∞ ₁ fn (x) dx = 0 because fn (x)0 uniformly. True 1 nx² +1' False…

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Q: 25. Suppose f: (a, b) → R is differentiable and f'(x)| ≤ M for all x E (a, b). Prove that f is…

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Q: Let Y = {−1, 1}. Then, (1) There is no continuous function f : X → Y , where Y is equipped with…

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Q: 5. If f (2) = 5 for the continuous function f (x), calculate the following limit value | F(t) dt lim…

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Q: Suppose f(z) is defined on A = {z Re(2) >0} such that f(z) is differentiable at each point in A and…

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Q: Prove the following. If f, g: [a, b] → R are continuous and both are differentiable on the open…

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Q: Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

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