Elementary Analysis (Advanced Math)

Elementary Analysis (Advanced Math)

Last six digits of UID:

1. (15 points) Let f(x) be a strictly increasing, continous function on R. Let S be a subset of R. The image f(S) is a subset of R. Prove that,

sup f(S) = f(sup S).

Hint: There is a sequence tends to sup S. Continuity means f and lim commute with each other.

 

no astocoeee

 

 

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2. (15 points) Consider the following function:

f(x) =

( 1 if x = 0; 1 x2 sin (x

2 ) if x ”= 0;

.

Prove that f(x) is uniformally continous on R. (Hint: You can try to prove f Õ(x) is bounded. The derivative f Õ(0) is not easy to calculate, therefore you can consider divide the interval into three parts. If you need more hints, check textbook examples.)

mm naorotaroree

 

 

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3. (15 points) Let f(x) be a function satisfies the following condition:

|f(x) ≠ f(y)| Æ sin2(x ≠ y). Prove that f(x) is a constant function.

Hint: Try prove the derivative is always zero.

mu Rnnttee

 

 

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4. (15 points) Calculate the Darboux Upper Integration U(f) and Lower Integration L(f) of the following function on the interval [≠1, 1].

f(x) =

( x if x Ø 0; ≠x if x < 0;

.

Hint: Check the last lecture notes/video.

season ending

 

 

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5. (10 points) Let u(x) be a continous function defined on R and u(0) = 0. We consider a function f(x) which is di?erentiable at x = 0. Prove that

lim xæ0

f(u(x)) ≠ f(0) u(x)

= f Õ(0).

(Hint: Use ‘ ≠ ” language.)

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6. (10 points) Calculate

lim næŒ

1

n (n!)

1 n .

Check textbook Section 12 for the result about limsup and liminf.

mn

brooch

 

 

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7. (10 points) (Advanced Topic I: Analysis and Fixed Point Theory).

Definition: A point xú œ R is a fixed point of f , if and only if f(xú) = xú. Question: Let f(x) : [0, 1] æ [0, 1] be a continous function. Prove that, f(x) has a fixed point.

Hint: Prove by contradiction. Assume that for any point x œ [0, 1], f(x) ”= x. Then check the boundary f(0) and f(1). Apply Intermediate Value Theorem.

Remark: There are some fixed point-related questions in the textbook: e.g. 29.18 (a special case of the Banach fixed-point theorem.). This

question is a special case for Brauwer’s Fixed Point Theorem. You don’t

have to follow the hint, there are quite a lot of di?erent proofs for this

question.

darn instantaneous

 

 

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8. (10 points) (Advanced Topic II: Analysis and Measure)

Definition: We say that a set S µ R has 0-measure if and only if, for any ‘ > 0 there exists a sequence of close intervals Ik = [lk, rk], k = 0, 1, 2, · · · , such that S µ

S+Πk=0 Ik and the total length of all intervals

P+Œ k=0(rk ≠ lk) <

‘.

Question: Prove that Q is a 0-measure set in R. Hint: Check the video/lectures about rational numbers. One can enumerate rational numbers. For each rational number qk, you can find a close interval Ik cover it. To control the total length of all interval Ik, you can set the length of Ik+1 equal to 12 of Ik. Then you can find a suitable length for the interval I0 to make the total length less than ‘.

Remark: Measure theory is a fundamental tool for propability theory. If we define function on [0, 1]

f(x) =

( 1 if x œ Q; 0 if x /œ Q;

.

This function is Not Riemann Integrable. However, as the measure of Q is 0, this function is Lebesgue Integrable.

mm tensions