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Questions related from Dinu Teodorescu
Can you provide an example of unbounded function f:R->R, discontinuous at every point x belongs R, satisfying the following property: f(xn) converges for all real numbers convergent sequence (xn)n?
17 September 2021 2,783 20 View
Let H be an infinite dimensional non-separable real Hilbert space and T:H->H be a linear and bounded operator. Let s be an eigenvalue of T. The set Es(T)={x/T(x)=sx} is a closed subspace of H. In...
04 June 2021 6,219 5 View
Let E be a real Banach space and E' its dual. Let f:E->R and a belongs to E. We say that f is differentiable at the point a if it exists Df(a) belongs to E' so...
01 January 1970 4,453 7 View
Let f:R->R be a continuous function such that f(xsin(1/x))=f(1/x)*cos(1/x) for all x in R\{0}. Prove that f(0)=f(1) and the equation f(x)=0 has at least one solution in every closed interval...
01 January 1970 5,387 6 View
Let ABC be a triangle with angle BAC
01 January 1970 818 8 View
Let f:R->R be a continuous real function and an=(n/(n+f(1/2)+f(1/3)+...+f(1/n)))n. Find all f(0) whichfor (an) is a convergent sequence and establish all possible limits of (an)!
01 January 1970 1,349 11 View
In attach you can find a problem( problem 4, page 73) from the book Elements d'Analyse, Tome 1 (Gauthier-Villars, 1968) of Jean Dieudonne. At the end of each section in his books from collection...
01 January 1970 5,692 8 View
If f is a continuous function from R to R having a periodic antiderivative ( primitive ), then the set of its zeros is at least countable and contains an unbounded sequence (xn) such that f(xn/n)...
01 January 1970 7,921 5 View
Let n>2 be a natural number and a=cos(2pi/n) + isin(2pi/n). Let i,j,k be three distinct elements of set {0,1,2,...,n-1}. Find all complex numbers z satisfying /z-a^i/=/z-a^j/=/z-a^k/, where /u/...
01 January 1970 7,314 4 View
It exist functions f:R->R with f(R)=R, satisfying: [ at every point x from R f has a finite limit l(x), l(x) being in R-{f(x)} for all x in R ] ? If yes, can such function have the intermediate...
01 January 1970 8,130 6 View
Let f:I->R be a C1 ( continuously differentiable) real function, where I=[0,infinity), f-f' having finite limit at infinity. Prove that there are two constants a>0, b>=0 such that...
01 January 1970 119 12 View
We denote as Re(v) and Im(v) the real, respectively the imaginary part of complex number v. Let z be a complex number satisfying /z/
01 January 1970 4,394 4 View
E(x)=(sinx/sqrt(x))+(cosx/sqrt(x+1)) . Prove that modulus of E(x) 0.
01 January 1970 2,423 9 View
O is a point in the interior of triangle ABC and M, N, P are the othogonal projections of O on BC, CA, respectively AB. Prove that the perpendicular from A to PN, the perpendicular from B to PM...
01 January 1970 8,639 7 View
One hundred circles all share the same centre, and they are named C1, C2, C3, and so on up to C100. For each whole number n between 1 and 99 inclusive, a tangent to circle Cn crosses circle Cn+1...
01 January 1970 3,081 1 View
Using the relation between (integral from 0 to 1)(xf'(x)f(x))dx and f^2(1), where f is a C^1 real function on [0,1], proof that (integral from 0 to 1)( x^2)(e^(2x^2)dx < (3e^2-5)/12. f' is the...
01 January 1970 5,009 3 View
Let (xn) be a real numbers sequence so that the sequence nxn+2^(xn) converges to a limit denoted by x. Prove that the sequence nxn+a^(xn) converges also to x, for all real number a>0.
01 January 1970 8,767 9 View
