Write ‘T' for true or ‘F' for false. You do not need to show any work...
true or false The real valued function f : (1,7) + R defined by f(x) = 2is uniformly contin- uous on (0,7). Let an = 1 -1/n for all n € N. Then for all e > 0) and any N E N we have that Jan - am) < e for all n, m > N. Let f :(a,b) → R be a differentiable function, if f'() = 0 for some point Xo € (a, b) then X, is...
Suppose fon (0,1) is uniformly continuous. Show that there is a real number A such that the function F defined by F(O)=A, F(x)=f(x) if x € (0,1), is continuous on (0,1]. (Suggestion: Show first that if {Xn}, Xn € (0,1] has lim xn = 0, then {f(xn)} is a Cauchy no sequence. Then show this sequence has the same limit no matter which {Xn} sequence going to you choose).
With justification in each one. Clarification; why if true and why if false? Please Determine whether the following statement is true or false: • Iff: R+R is differentiable and strictly increasing on R, then f'(1) > 0 VI ER • If S: R R is continuous and f(x) - ron Q, then (V3) - 3. • If f,g: (0,1) - Rare functions such that \S(1)-f(y) = g(1)-9(y) for all 1, y € (0, 1) and g is continuous on (0,1),...
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
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...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
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...
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...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...