Record some Problems

May be used in class of analysis:

Problem. Suppose $f'(x)<0<f''(x)$ for $x<a$; and $f'(x)>0>f''(x)$ for $x>a$. Prove that $f$ is not differentiable at $a$.  

Problem. If $f:\R\to \C^\times$ is a group homomorphism and is continuous, prove that $f(x)=e^{cx}$ for some $c\in \C$.

Problem. Let $C[a,b]$ be the ring of continuous functions on the finite closed interval $[a,b]\subseteq \R$.
(a) For $c\in [a,b]$, prove that $I_c$ given by $$I_c=\{f\in C[a,b]:f(c)=0\}$$ is a maximal ideal of $c[a,b]$.
(b) Prove that every maximal ideal of $C[a,b]$ is $I_c$ for a unique $c\in [a,b]$.

Problem. Define a sequence by $a_1 = 1$, $a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$, if it exists.

Problem. Denote $A = \{ x \in \mathbb{R} : \lim_{n\to\infty} \{ 2^n x\} = 0 \}$. Prove that $m(A) = 0$. Is $A$ countable or uncountable?

Notation: $\{x\}$ denotes the fractional part of $x$, given by $\{x\} = x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the unique integer for which $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$.

Problem. Find $\dis  \prod_{k=0}^\infty(e^{a\cdot 2^{-k}}+e^{-a \cdot 2^{-k}}-1)$.

Problem. Find $\dis \lim_{n\to\infty}\left(\prod_{k=1}^{n}\cos{\frac{\pi k}{2n}}\right)^{\frac{1}{n}}$.

Problem.  Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable ($C^1$) function such that $\dis f(x) = f(\frac{x}{2})+ \frac{x}{2} f'(x)$.

Show that $f(x)$ is a linear function, i.e. $\exists (a,b) \in \mathbb{R}^2$, $f(x)= ax+b$.

Problem. Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that $$\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0.$$ Then prove that $$\lim_{N\to\infty}\int_0^\pi\frac{\sin(N+\frac12)x}xdx=\pi/2.$$ Finally, prove that $$\int_0^\infty\frac{\sin x}xdx=\pi/2.$$

Problem. Let $a>0$ and $\{x_n\}$ a bounded sequence of real numbers. If $\{x_n\}$ doesn't converge and $\lim_{n \to \infty} (x_{n+1}+ax_n)=0$, prove that $a=1$.
Remark. We can generalize the problem by merely assuming $\lim_{n \to \infty} (x_{n+1}+ax_n)$ exists.

Problem. Let $f: \mathbb{R} \to (0,\infty)$ be a continuous differentiable function such that $f'(x)=f(f(x))$ for all $x\in \mathbb{R}$. Show that there is no such function. 

Problem. Suppose $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that, $|f(x)-f(y)| \ge \frac{1}{2}|x-y|$ for all $x,y \in \mathbb{R}$. Is $f$ necessarily one to one and onto?

Problem. Let $f:[0,1]\to \R$ be continuous and differentiable on $(0,1)$ such that $f(0)>0,f(\frac{1}{2})<0$ and $f(1)>1$. Show that there is a $\xi\in (0,1)$ such that $f'(\xi)=f(\xi)$.