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Real Analysis
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...
Real Analysis
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.)
Real Analysis
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.
Real Analysis
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.
Real Analysis
1/2 is a limit of the interval] - 2.5,1.5[, true or false
Real Analysis
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.
f(x)= (7/2)-2x, ∀ x∈[1,3]
Is f integrable over the interval [1,3]? Justify.
Real Analysis
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
Real Analysis
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]
Real Analysis
true/false? prove.
1/2 is a limit of the interval ]− 2.5,1.5[
1/2 is a limit of the interval ]− 2.5,1.5[
Real Analysis
If a ∈ R is such that o , ≤ a ≤ ε ∀ ε > 0 ,then show that a = 0.
