### Archive

Posts Tagged ‘continuity’

## Putnam 2017 A3 – Solution

Problem A3. Denote ${\phi = f-g}$. Then ${\phi}$ is continuous and ${\int_a^b \phi = 0}$. We can see that

$\displaystyle I_{n+1}-I_n = \int_a^b (f/g)^n \phi = \int_{\phi\geq 0} (f/g)^n \phi+ \int_{\phi<0} (f/g)^n \phi$

Now note that on ${\{ \phi>=0\}}$ we have ${f/g \geq 1}$ so ${(f/g)^n \phi \geq \phi}$. Furthermore, on ${\{\phi<0\}}$ we have ${(f/g)^n <1}$ so multiplying with ${\phi<0}$ we get ${(f/g)^n \phi \geq \phi}$. Therefore

$\displaystyle I_{n+1}-I_n \geq \int_{\phi \geq 0} \phi + \int_{\phi<0} \phi = \int \phi = 0.$

To prove that ${I_n}$ goes to ${+\infty}$ we can still work with ${I_{n+1}-I_n}$. Note that the negative part cannot get too big:

$\displaystyle \left|\int_{ \phi <0 } (f/g)^n \phi \right| \leq \int_{\phi<0} |\phi| \leq \int_a^b |f-g|.$

As for the positive part, taking ${0<\varepsilon< \max_{[a,b]} \phi}$ we have

$\displaystyle \int_{\phi\geq 0} (f/g)^n \phi \geq \int_{\phi>\varepsilon}(f/g)^n \varepsilon.$

Next, note that on ${\{ \phi \geq \varepsilon\}}$

$\displaystyle \frac{f}{g} = \frac{g+\phi}{g} \geq \frac{g+ \varepsilon}{g}.$

We would need that the last term be larger than ${1+\delta}$. This is equivalent to ${g\delta <\varepsilon}$. Since ${g}$ is continuous on ${[a,b]}$, it is bounded above, so some delta small enough exists in order for this to work.

Grouping all of the above we get that

$\displaystyle I_{n+1}-I_n \geq \int_{\phi \geq 0} (f/g)^n \phi \geq \int_{\phi>\varepsilon} \varepsilon (1+\delta)^n.$

Since ${|\phi>\varepsilon|>0}$ we get that ${I_{n+1}-I_n}$ goes to ${+\infty}$.

## SEEMOUS 2014

Problem 1. Let ${n}$ be a nonzero natural number and ${f:\Bbb{R} \rightarrow \Bbb{R}\setminus \{0\}}$ be a function such that ${f(2014)=1-f(2013)}$. Let ${x_1,..,x_n}$ be distinct real numbers. If

$\displaystyle \left| \begin{matrix} 1+f(x_1)& f(x_2)&f(x_3) & \cdots & f(x_n) \\ f(x_1) & 1+f(x_2) & f(x_3) & \cdots & f(x_n)\\ f(x_1) & f(x_2) &1+f(x_3) & \cdots & f(x_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ f(x_1)& f(x_2) & f(x_3) & \cdots & 1+f(x_n) \end{matrix} \right|=0$

prove that ${f}$ is not continuous.

Problem 2. Consider the sequence ${(x_n)}$ given by

$\displaystyle x_1=2,\ \ x_{n+1}= \frac{x_n+1+\sqrt{x_n^2+2x_n+5}}{2},\ n \geq 2.$

Prove that the sequence ${y_n = \displaystyle \sum_{k=1}^n \frac{1}{x_k^2-1} ,\ n \geq 1}$ is convergent and find its limit.

Problem 3. Let ${A \in \mathcal{M}_n (\Bbb{C})}$ and ${a \in \Bbb{C},\ a \neq 0}$ such that ${A-A^* =2aI_n}$, where ${A^* = (\overline A)^t}$ and ${\overline A}$ is the conjugate matrix of ${A}$.

(a) Show that ${|\det(A)| \geq |a|^n}$.

(b) Show that if ${|\det(A)|=|a|^n}$ then ${A=aI_n}$.

Problem 4. a) Prove that ${\displaystyle \lim_{n \rightarrow \infty} n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx=\frac{\pi}{2}}$.

b) Find the limit ${\displaystyle \lim_{n \rightarrow \infty} n\left(n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx-\frac{\pi}{2} \right)}$

## Asymptotic characterization in terms of sequence limits

Suppose ${f:(0,\infty) \rightarrow \Bbb{R}}$ is a continuous function such that for every ${x>0}$ we have

$\displaystyle \lim_{n \rightarrow \infty} f(nx)=0.$

Prove that ${\lim\limits_{x \rightarrow \infty} f(x)=0}$.

## Miklos Schweitzer 2013 Problem 7

November 22, 2013 1 comment

Problem 7. Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function (that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}}$) for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous.

Miklos Schweitzer 2013

Categories: Algebra, Analysis, Geometry

## Continuity in Geometry

Here are a few interesting geometry problems which use continuity problems in their solutions.

Pb 1. Consider three parallel lines in the plane ${d_1,d_2,d_3}$. Prove that there exist points ${A_i\in d_i}$ such that the triangle ${A_1A_2A_3}$ is equilateral.

Pb 2. Consider a triangle ${ABC}$. Prove that ${ABC}$ is equilateral if and only if for every point ${M}$ in the plane we can construct a triangle with sides ${MA,MB,MC}$.

## Agreg 2012 Analysis Part 4

A Fixed Point Theorem

This parts wishes to extend the next result (which can be used without proof) to infinite dimension.

Theorem. (Browuer) Consider ${F}$ a finite dimensional normed vector space. Consider ${C\subset F}$ a convex, closed, bounded non-void set. If ${f:C\rightarrow C}$ is a continuous application, then ${f}$ has a fixed point in ${C}$.

1. In ${\ell^2(\Bbb{N})}$ endowed with ${\|\cdot \|_2}$ we consider the following application:

$\displaystyle f: B(0,1) \subset \ell^2(\Bbb{N}) \rightarrow \ell^2(\Bbb{N})$

$\displaystyle f(x) =(\sqrt{1-\|x\|_2},x_0,x_1,...).$

Prove that ${f}$ is continuous with values in the unit sphere of ${\ell^2(\Bbb{N})}$, but ${f}$ does not admit any fixed points.

2. Consider ${E}$ a normed vector space, ${B}$ a closed, bounded non-void subset of ${E}$ and ${f:B \rightarrow E}$ a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)

i) Let ${n \in \Bbb{N}\setminus \{0\}}$. We can cover ${\overline{f(B)}}$ (which is compact) by a finite number ${N_n}$ of open balls of radius ${\frac{1}{n}}$: ${\overline{f(B)} \subset \displaystyle \bigcup_{i=1}^{N_n} \mathring{B}(y_i,\frac{1}{n})}$ with ${y_i \in \overline{f(B)}}$ for every ${i}$. For ${y \in E}$ we define

$\displaystyle \psi(y) =\begin{cases} \frac{1}{n}-\|y-y_i\| & \text{ if } y \in B(y_i,\frac{1}{n}) \\ 0 & \text{ otherwise} \end{cases}$

Prove that ${\Psi : y \in \overline{f(B)} \mapsto \sum_{i=1}^{N_n} \psi_i(y)}$ is continuous and that there exists ${\delta>0}$ such that for ${y \in \overline{f(B)}}$ we have ${\Psi(y)\geq \delta}$.

ii) We introduce the application ${f_n : B \rightarrow E}$ defined by

$\displaystyle f_n(x)= \left( \sum_{i=1}^{N_n} \psi_i(f(x))\right)^{-1} \sum_{i=1}^{N_n} \psi_i(f(x))y_i.$

Prove that for every ${x \in B}$ we have ${\|f(x)-f_n(x) \|\leq \frac{1}{n}}$.

Categories: Analysis, Fixed Point