Posts Tagged ‘continuity’

Putnam 2017 A3 – Solution

December 4, 2017 Leave a comment

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}.



March 18, 2014 Leave a comment

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

December 17, 2013 Leave a comment

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}.

Read more…

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

Read more…

Continuity in Geometry

November 22, 2013 Leave a comment

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}.

Read more…

Agreg 2012 Analysis Part 4

October 20, 2013 Leave a comment

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}}.

Read more…

Arcs on a circle

February 3, 2012 Leave a comment

Suppose that we have a finite set of arcs on a circle, with the property that every two of them intersect. Prove that there exists a diameter which intersects all arcs.

Read more…

%d bloggers like this: