(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|&l...
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...
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...
(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
10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous function on [-MI, M]. Show that g(Um(x)) uniformly to g(f(r)) on E 10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a pair of measurable functions such that both are integrable on a set A E A and g(x) < fn(x)<h(x), for all E A and ne N. Prove that / lim sup fn du fn dulim sup fn du lim inf fn du lim inf n o0 A n-oo A noo n00...
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
3. Recall that R([0, 1]) is the normed linear space of integrable functions, with norm 1/2 Ils le = (150)Par)". Let (fn)nen be a sequence of functions in R, defined by 1<3 fn(x) = 1 VI V 0 < (a) Prove that (fn)nen is Cauchy. (b) Prove that (fn) does not converge in R([0, 1]). (Note: If it did, then what must the limit function be? Can this candidate function be in R?)
(10) Prove that if (fn) is a sequence of uniformly continuous functions on the interval (a, b) such (a, b), then f is also uniformly continuous on (a, b) that f funiformly on en dr 0. (11) Show that lim n-+o0 e (10) A G.. 11d
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.
Tamo . Suppose that a sequence of functions fn converges pointwise to a function f on a set E, but there exists a sequence of points In E E such that \fn(2n) – f(2n) > for some strictly positive l. Then fn does not converge uniformly to f on E. (You don't need to prove this here, but it should be clear why this is true.) Now let nar2 fn(L) = 2 +n323 Show that fn converges pointwise on [0,0]...