1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l if z e R is rational, if zER is irrational f(z) Show that limfr) does not exists for any ceR. 4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such...
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
5. Define f:R + R by f(x) = x2 if x is rational, and f(x) = 0 if x is irrational. Show that f'(0) exists and is equal to zero.
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Question 4 of the image Prove that, for all n 1 1 Arrange the following rational numbers in increasing order: (i) x, is a rational number 61/99, 3/5, 17/30, 601/999, 599/1001. g 0 2 Find positive integers r and s such that r/s is equal to the repeating decimal (ii) 2 x5/2. Find an expression for x - 5 involving x,-5, and hence explain (without formal proof) why x, tends to a limit which is not a rational number 0.30024....
s y cosci) it x80 Let fix,y) = {'o it X=0 know that limit of fix,y) is o as (x, y)+10.0) lingo I liga flxy) 18 limon culing flXy)] Explain why this does not contradict following Theorem and Prove your explanation: Suppose that I and I are open intervals, that a e I and be J, and that f : (1 x J) \ {(a, b)} → R. If g(x):= lim f(x, y) y → exists for each xe I...
Let r be any rational number and define L = { x in Q: x < r }, the set of rational numbers less than r. Show that L is a Dedekind cut by proving the following properties: A. There exists a rational number x in L and there exists a rational number y not in L. ( This proves L is nonempty and L is not equal to Q) B. If x in L, then there exists z in...
gol The fixed-point iteration Pn+1 = g(P) converges to a fixed point p = 0 of g(x) = x for all 0 < po < 1. The order of convergence of the sequence {n} is a > 0 if there exists > O such that lim Pn+1-pl =X. -00 P -plº Use the definition (6) to find the order of convergence of the sequence in (5).