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(Solved): (Integration) Let a,binR with g:[a,b]->Rga. Let g:[a,b]->R be decreasing. Prove that g is Riem ...
(Integration) Let
a,binRwith
g:[a,b]->Rga. Let g:[a,b]->R be decreasing. Prove that g is Riemann integerable.425 For problems 1-5 you chose one and I chose one. For problem 6, one part is chosen at random. You will have a whiteboard available to demonstrate your argument, feel free to use pictures. (Completeness) Prove that
Ris complete using the tools from this class. (Compactness) Let
x,Ybe metric spaces. Let
Asubexbe nonempty and compact, let
f:A->Y. Show that if
fis continuous then
f(A)is compact. (Connectedness) Let
xa be metric space. Let
A,Bsubexbe nonempty and connected. Is
A\cup Bconnected? Prove it or give a counterexample. (Counterexample must be specific!) (Differentiation) Let
a,binRwith
f:[a,b]->Rcin(a,b)cca,binRf:[a,b]->Rf^(')xin(a,b)f^(')a,binRg:[a,b]->RgRRA\cap Bf(x)=(1)/(x)f(1)=1,f(-1)=-1cf(c)=0cin(a,b)cca. Let g:[a,b]->R be decreasing. Prove that g is Riemann integerable.
Chosen at random.
(a) Give an example of a noncomplete subset of R and show why it is not complete. Be specific.
(b) Explain the min-max theorem. Then give an example of a bounded noncompact subset of R to for which the an min or max is not achieved.
(c) Is A\cap B connected? Prove it or give a counterexample. (Counterexample must be specific!)
(d) Explain why the function f(x)=(1)/(x) is continuous, f(1)=1,f(-1)=-1, yet there is no value c such that f(c)=0. Explain why this doesn't contradict the intermediate value theorem.
(e) Let cin(a,b). If a function is continuous at c it is differentiable at c. Prove it or give a counterexample. (Counterexample must be specific!)a. Let f:[a,b]->R be differentiable. Suppose that the left-hand limits and right-hand limits of f^(') exist at all xin(a,b). Prove that f^(') is continuous. (There are several cases, choose just one case, illustrate with a picture and an argument.)a. Let f:[a,b]->R. (One of the two.)
(a) Let cin(a,b). If a function is differentiable at c it is continuous at c. Prove it or give a counterexample. (Counterexample must be specific!)
(b) Let a,binR with a. Let f:[a,b]->R be differentiable. Suppose that the left-hand limits and right-hand limits of f^(') exist at all xin(a,b). Prove that f^(') is continuous. (There are several cases, choose just one case, illustrate with a picture and an argument.)