Find all bijective (1-1 & on to) and continuous functions \( h: \mathbb{R}\longrightarrow\mathbb{R}\), such that:
(a) there is \( x_0\in\mathbb{R}\), such that \( h(x_0)=x_0\) and
(b) \(h(x)+h^{-1}(x)=2x\,,\) for all \(x\in\mathbb{R}\) .
Search found 308 matches
- Tue Nov 10, 2015 11:39 am
- Forum: Real Analysis
- Topic: functional equation (01)
- Replies: 1
- Views: 2244
- Tue Nov 10, 2015 11:35 am
- Forum: Real Analysis
- Topic: Sequence (01)
- Replies: 1
- Views: 2294
Sequence (01)
Let \(\{{a_{n}}\}_{n=1}^{\infty}\) a sequence of real numbers such that \[0<a_{1}<a_2\quad {\text{ and}}\quad a_{n+1}=\sqrt{a_{n}\,a_{n-1}}\, , \;n\geqslant2\, .\] a) Prove that the sequence \(\{{a_{n}}\}_{n=1}^{\infty}\) converges. b) Prove that \(\displaystyle\mathop{\lim}\limits_{n\rightarrow{+\i...
- Tue Nov 10, 2015 11:30 am
- Forum: Real Analysis
- Topic: is contractive?
- Replies: 0
- Views: 1947
is contractive?
A sequence \(\{{\alpha_{n}}\}_{n=1}^{\infty}\) is contractive iff there exists a constant \(c\), with \(0<c<1\), such that, for all \(n\in\mathbb{N}\), holds: \[|a_{n+2}-a_{n+1}|\leqslant c\,|a_{n+1}-a_{n}|\] Examine if the sequence \[a_{n}=({\underbrace{\sin\circ\sin\circ\ldots\circ\sin}_{n-{\rm{ti...
- Tue Nov 10, 2015 11:27 am
- Forum: Projective Geometry, Solid Geometry
- Topic: Inscribed sphere of rhombic triacontahedron
- Replies: 1
- Views: 4933
Re: Inscribed sphere of rhombic triacontahedron
A rhombic triacontahedron has \(30\) faces, all of which are golden rhombi. A golden rhombus is a rhombus such that the ratio of the long diagonal \(\varDelta\) to the short diagonal \(\delta\) is equal to the golden ratio \(\Phi\), ie \[\frac{\varDelta}{\delta}=\Phi=\frac{1+\sqrt{5}}{2}\quad(1)\,.\...
- Mon Nov 09, 2015 4:23 am
- Forum: Real Analysis
- Topic: Does such function exist?
- Replies: 1
- Views: 2510
Does such function exist?
Does exist a real function \(f:[a,b]\longrightarrow{\mathbb{R}}\) which is bounded, monotonic and discontinuous in uncountable many points of \([a,b]\) ?
- Mon Nov 09, 2015 4:20 am
- Forum: Linear Algebra
- Topic: Fibonacci numbers as determinants
- Replies: 1
- Views: 2440
Fibonacci numbers as determinants
Let \(\{{F_{n}}\}_{n=1}^{\infty}\) be the Fibonacci sequence defined as \[F_{n}=F_{n-1}+F_{n-2}\,,\;n\geqslant3\,, \quad F_1=F_2=1\,.\] Prove for every \(n\in\mathbb{N},\,n\geqslant2\) that the \(n\)-th term \(F_n\) of the Fibonacci sequence it is given by the determinant of the \((n-1)\times(n-1)\)...
- Mon Nov 09, 2015 1:47 am
- Forum: Projective Geometry, Solid Geometry
- Topic: Inscribed sphere of rhombic triacontahedron
- Replies: 1
- Views: 4933
Inscribed sphere of rhombic triacontahedron
Consider a rhombic triacontahedron \(R\) with edge length \(1\) and the inscribed sphere \(S\) of \(R\) (tangent to each of the rhombic triacontahedron's faces). Prove that the radius \(r\) of \(S\) has length \[r=\frac{\Phi^2}{\sqrt{1 + \Phi^2}} =\frac{3 + \sqrt{5}}{\sqrt{10 + 2\sqrt{5}}}\,,\] wher...
- Mon Nov 09, 2015 1:40 am
- Forum: Differential Geometry
- Topic: Circle or line
- Replies: 0
- Views: 1950
Circle or line
Let \(\overrightarrow{r}(s)\) a natural parametrization of a plane curve. If every tangent line of the curve has the same (constant) distance from a fixed point, prove that the curve must be either a part of a circle, or a part of a line.