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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.

1/2 is a limit of the interval] - 2.5,1.5[, true or false

Find the upper and lower integrals of the function f defined by

f(x)= (7/2)-2x, ∀ x∈[1,3]

Is f integrable over the interval [1,3]? Justify.

Let b = supS where S is a bounded subset of R.

(a) Given epsilon > 0, show that there exists an s in S with b - (epsilon) ≤ s ≤ b

(a) Given epsilon > 0, show that there exists an s in S with b - (epsilon) ≤ s ≤ b

true/false? prove..

Every function differentiable on [a, b] is bounded on [a, b]

Every function differentiable on [a, b] is bounded on [a, b]