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With justification in each one. Clarification; why if true and why if false? Please Determine whether...
Write ‘T' for true or ‘F' for false. You do not need to show any work or justify your answers for this question. The questions are 2 points each. (a) __If (xn) is a convergent sequence (converging to a finite limit) and f:RR is a continuous function, then (f (xn)) is a convergent sequence. (b) _If (xn) is a Cauchy sequence with Yn € (0,1) and f :(0,1) + R is contin- uous, then (f(xn)) is also a Cauchy sequence....
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers. (a) Suppose both f and g are continuous on (a, b) with f > 9. If Sf()dx = Sº g(x)dx, then f(x) = g(x) for all 3 € [a, b]. (b) If f is an infinitely differentiable function on R with f(n)(0) = 0 for all n = 0,1,2,..., then f(x) = 0 for all I ER. (c) f is improperly integrable on (a,...
Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-),...
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...
For each of the following statements, either prove it is true, or provide a counterexample to show that it is false. (a) If (sn) is a sequence such that lim sn = 0, then lim inf|sn= 0. (b) If f : [0, 1] + R is a function with f(0) < 0 and f(1) > 0, then there exists CE (0,1) such that f(c) = 0. (c) If I is an interval, f:I + R is continuous on I, and...
Please let me know whether true or false If false, please give me the counter example! (a) If a seriesE1an converges, then lim,n-0 an = 0. m=1 (b) If f O(g), then f(x) < g(x) for all sufficiently large . R is any one-to-one differentiable function, then f-1 is (c) If f: R differentiable on R (d) The sequence a1, a2, a3, -.. defined by max{ sin 1, sin 2,-.- , sin n} an converges (e) If a power series...
state wether true or false. no need to justify (a) If a sequence is divergent, then it contains two convergent subsequences with distinct limits. (b) If a function f is uniformly continuous on all real numbers, the f is continuous at each b in all real numbers. (c) If a function f^2 is continuous, them so is f continuous. (d) If a differentiable function f on all real numbers satisfies f(0)=-1 and f'(y)>0 for all y>=0, then there is a...
#55, 59 In Exercises 55 and 56, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. 55. If f is continuous at x = a, then f is differentiable at x = a. 56. If f is continuous at x = a and g is differentiable at x = a, then lim f(x)g(x) = f(a)g(a). X 57. Sketch the...
math analysis 1. decide which of the following statements ore true os false. Prove the true ones and give a counter example for the false ones a) If f and I are continuous on 19.6], frane glas and f(b) > g(6), then there is a cca.bI such that fcc) - grey b) suppose that fandy are defined and finile Volved an open inken val I which contains a, that fis cóntinuous at a, and that fla) & 0. Then g...
Please answer c d e 3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...