Ross 34.2, 34.5, 34.7

Optional:

Rudin: Ex 15 (Hint: use 10( c ) ), 16

and an extra one:

Let $f:[0,1] \to \R$ be given by $$ f(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sin(1/x) &\text{if } x \in (0,1] \end{cases}. $$ And let $\alpha: [0, 1] \to \R$ be given by $$ \alpha(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sum_{n \in \N, 1/n<x} 2^{-n} &\text{if } x \in (0,1] \end{cases}. $$ Prove that $f$ is integrable with respect to $\alpha$ on $[0,1]$. Hint: prove that $\alpha(x)$ is continuous at $x=0$.

Ross 33.4, 33.7, 33.13, 35.4, 35.9(a)

The last two weeks, we studied derivation, with topics like

• mean value theorem
• intermediate value theorem
• Taylor expansion

Exercises:

• Read Ross p257, Example 3 about smooth interpolation between $0$ for $x \leq 0$ and $e^{-1/x}$ for $x>0$. Construct a smooth function $f: \R \to \R$ such that $f(x)=0$ for $x\leq 0$ and $f(x)=1$ for $x\geq 1$, and $f(x) \in [0,1]$ when $x \in (0,1)$.
• Rudin Ch 5, Ex 4 (hint: apply Rolle mean value theorem to the primitive)
• Rudin Ch 5, Ex 8 (ignore the part about vector valued function. Hint, use mean value theorem to replace the difference quotient by a differential)
• Rudin Ch 5, Ex 18 (alternative form for Taylor theorem)
• Rudin Ch 5, Ex 22

Let's consider a few examples of sequences and series of functions.

1. Let $f_n(x) = \frac{n + \sin x} {2n + \cos n^2 x}$, show that $f_n$ converges uniformly on $\R$.

2. Let $f(x) = \sum_{n=1}^\infty a_n x^n $. Show that the series is continuous on $[-1, 1]$ if $\sum_n |a_n| < \infty$. Prove that $\sum_{n=1}^\infty n^{-2} x^n$ is continuous on $[-1, 1]$.

(In general, if one only know that $\sum_n a_n$ and $\sum_n (-1)^n a_n$ converge, then the result still holds, but is harder to prove. See Ross Thm 26.6)

3. Show that $f(x) = \sum_n x^n$ represent a continuous function on $(-1,1)$, but the convergence is not uniform. (Hint: to show that $f(x)$ on $(-1,1)$ is continuous, you only need to show that for any $0<a<1$, we have uniform convergence on $[-a, a]$. Use Weierstrass M-test. )

This week we proved the equivalence of the two notions of compactness. Here are some more problems

1. If $X$ and $Y$ are open cover compact, can you prove that $X \times Y$ is open cover compact? (try to do it directly, without using the equivalence between open cover compact and sequential compact)

2. Let $f: X \to Y$ be a continuous map between metric spaces. Let $A \In X$ be a subset. Decide if the followings are true or not. If true, give an argument, if false, give a counter-example.

• if $A$ is open, then $f(A)$ is open
• if $A$ is closed, then $f(A)$ is closed.
• if $A$ is bounded, then $f(A)$ is bounded.
• if $A$ is compact, then $f(A)$ is compact.
• if $A$ is connected, then $f(A)$ is connected.

3. Prove that, there is not continuous map $f: [0,1] \to \R$, such that $f$ is surjective. (there is a surjective map from $(0,1) \to \R$ though)