Real Analysis Answers

Suppose f is a function such that
f(x) = { x^2, x element Q
0 x element R - Q

Is f continuous at c=0? is f differentiable at c = 0?

assume that lim[1+2(-1)^n]Xn = 0. Prove that lim Xn exists and find it.

show that a nonempty finite set s subsets R contains its supremum

prove that there exist real numbers which is not algebraic

Let f(x) = |x|^1/2. Is f differentiable at c = 0?

Which of the following statements are true and why?

1.Any continuous function from the open unit interval (0,1) to itself has a fixed point.

2.logx is uniformly continuous on (1/2,+∞) .

3.If A,B are closed subsets of [0,∞) , then A+B={x+y|x∈A,y∈B} is closed in [0,∞)
4.A bounded continuous function on R is uniformly continuous.

5.Suppose f n (x) is a sequence of continuous functions on the closed interval [0,1] converging to 0 pointwise. Then the integral ∫ 1 0 f n (x)dx converges to 0 .

Using the "E-N" definition of the limit, show that

lim n[(n^2+2)^(1/2) - n] = 1

Let R be reflexive and transitive relation on a set S. Then R intersect R inverse is a
a)reflexive but not transitive relation.
b)transitive but not reflexive relation.
c)symmetric but not reflexive and transitive relation.
d)equivalence relation.

Solve for the unknown variables:
(a) (a, b) + (3, 1) = (6, 7)
(b) (11, c) + (d, 6) = (-12, −3)
(c) (5, 5) + (e, 4) = (2, f)
(d) (8, −3) − (5, g) = (h, 2)

Let X and Y be metric spaces and let f:=X→Y be a mapping. Pick out the true statements:

a. if f is uniformly continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y ;

b. if X is complete and if f is continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y ;

c. if Y is complete and if f is continuous, then the image of every Cauchy sequence in X is a Cauchy sequence in Y