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Homework Answers
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5. Let fn(x) = x"/n on [0, 1]. Show that (fr)nen converges uniformly to a differentiable...
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges...
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx)...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that...
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that the function f(x) is also differentiable. (All of these conditions are assumed to be true on a bounded, closed interval [a, b].) Prove or disprove: lim as n goes to infinity fn'(x) = f'(x)
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m =...
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
5. Let f,lr)- x *a. Show that {h} converges uniformly to 0 on [0, a] for...
5. Let f,lr)- x *a. Show that {h} converges uniformly to 0 on [0, a] for any a, 0 < a < 1. b. Does {f,) converge uniformly on [0, 1]?
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded a...
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10...
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn} converges pointwise. fn. Does {6 fn} converge to (b) For each n EN compute (c) Can the convergence of {fn} to f be uniform?
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly...
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
5. Let f,lr)- x *a. Show that {h} converges uniformly to 0 on [0, a] for any a, 0 < a < 1. b. Does {f,) converge uniformly on [0, 1]?
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn} converges pointwise. fn. Does {6 fn} converge to (b) For each n EN compute (c) Can the convergence of {fn} to f be uniform?
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
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