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Q.Let D be a metric on X determined all constant k s.t

(i) kd is a metric

(ii)(k+d) is a metric

(i) kd is a metric

(ii)(k+d) is a metric

Q.Find all metric on a set X consisting of one point and consisting of two points.

Q. Prove that |d(x,z)-d(y,z)|≤d(x,y)

let f(x) =modulus x^3.Compute f'(x),f"(x),for all real x,and show that f"'(0) does not exist

Q. Show that the real line is a metric space.

Q. Is d(x, y) =√(│x-y│) a metric space?(solve it)

Consider the Cantor-Lebesgue function F : C → R, where C is the Cantor set.

a) Show that F is continuous and actually F can be extended to the whole R so that it is continuous (to do

this, note that if (a, b) is an interval from the complement of C, then F(a) = F(b)).

b) Using (a), show that if f : R → R is continuous function and A is a measurable subset of R, then f(A) may

not be measurable.

c) Show that the inverse image of a measurable set under a continuous function is not always measurable

(compare with Problem 2).

Hint: For (c), notice that F is increasing, thus the function F(x) +x is strictly increasing (and continuous).

Therefore, F(x) + x has a continuous inverse...

a) Show that F is continuous and actually F can be extended to the whole R so that it is continuous (to do

this, note that if (a, b) is an interval from the complement of C, then F(a) = F(b)).

b) Using (a), show that if f : R → R is continuous function and A is a measurable subset of R, then f(A) may

not be measurable.

c) Show that the inverse image of a measurable set under a continuous function is not always measurable

(compare with Problem 2).

Hint: For (c), notice that F is increasing, thus the function F(x) +x is strictly increasing (and continuous).

Therefore, F(x) + x has a continuous inverse...

Fixing x, show that the Taylor series of e

tx− t

2

2 is given by X∞

n=0

t

nHn(x). (Explain

also why the series converges.)

tx− t

2

2 is given by X∞

n=0

t

nHn(x). (Explain

also why the series converges.)

Show that if A, B ⊆ R^d are closed sets, then A + B is measurable (in fact A + B is an Fσ set). Show, however that A + B may not be closed.

Evaluate ∫∫∫z2dx dydz

S

,where S is the solid region between the spheres ρ =1 and

ρ = 2, by using spherical coordinates.

S

,where S is the solid region between the spheres ρ =1 and

ρ = 2, by using spherical coordinates.