MathDB

Characterization of monotone functions?

Source:

7/14/2020

Let be a function

f:(0,\infty )\longrightarrow\mathbb{R}

satisfying the following two properties:
\text{(i) } 2\lfloor x \rfloor \le f(x) \le 2 \lfloor x \rfloor +2, \forall x\in (0,\infty )

\text{(ii) } f\circ f

is monotone
Can

f

be non-monotone? Justify.

functionfloor functionmonotonyalgebra

Nice geometric combinatorics

Source:

7/14/2020

In the Euclidean plane, let be a point

O

and a finite set

\mathcal{M}

of points having at least two points. Prove that there exists a proper subset of

\mathcal{M},

namely

\mathcal{M}_0,

such that the following inequality is true:

\sum_{P\in \mathcal{M}_0} OP\ge \frac{1}{4}\sum_{Q\in\mathcal{M}} OQ

inequalitiescombinatoricsgeometry

A complex-numbers condition for some perpendicularities

Source:

7/14/2020

Let be a natural number

n\ge 2

and an imaginary number

z

having the property that

|z-1|=|z+1|\cdot\sqrt[n]{2} .

Denote with

A,B,C

the points in the Euclidean plane whose representation in the complex plane are the affixes of

z,\frac{1-\sqrt[n]{2}}{1+\sqrt[n]{2}} ,\frac{1+\sqrt[n]{2}}{1-\sqrt[n]{2}} ,

respectively. Prove that

AB

is perpendicular to

AC.

complex numbersimaginary numbersalgebrageometryanalytic geometryComplex Geometry

Beautiful problem about non-continuous primitivable functions

Source:

7/16/2020

Let be areal number

r,

a nonconstant and continuous function

f:\mathbb{R}\longrightarrow\mathbb{R}

with period

T

and

F

be its primitive having

F(0)=0.

Define the funtion

g:\mathbb{R}\longrightarrow\mathbb{R}

as

g(x)=\left\{\begin{matrix} f(1/x), & x\neq 0 \\ r, & x=0 \end{matrix}\right.

Prove that: a) the image of

f

is closed. b)

g

has the intermediate value property if and only if

r\in f\left(\mathbb{R}\right) .

c)

g

is primitivable if and only if

r=\frac{F(T)}{T} .

functionIVPDarbouxreal analysiscontinuityprimitivability

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