MathDB

Problem 3

Part of 2007 VJIMC

Problems(2)

limit of f(tx)/f(x) as x->infinity, f has a constant sign

Source: VJIMC 2007 1.3

6/24/2021
A function f:[0,∞)→R∖{0}f:[0,\infty)\to\mathbb R\setminus\{0\} is called slowly changing if for any t>1t>1 the limit lim⁡x→∞f(tx)f(x)\lim_{x\to\infty}\frac{f(tx)}{f(x)} exists and is equal to 11. Is it true that every slowly changing function has for sufficiently large xx a constant sign (i.e., is it true that for every slowly changing ff there exists an NN such that for every x,y>Nx,y>N we have f(x)f(y)>0f(x)f(y)>0?)
functionlimitsreal analysis
h:f(x+h)=f(x) for some x is lebesgue measurable

Source: VJIMC 2007 2.3

6/24/2021
Let f:[0,1]→Rf:[0,1]\to\mathbb R be a continuous function such that f(0)=f(1)=0f(0)=f(1)=0. Prove that the set A:={h∈[0,1]:f(x+h)=f(x) for some x∈[0,1]}A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}is Lebesgue measureable and has Lebesgue measure at least 12\frac12.
real analysisset theoryfunction