The function g:(0,infty)->R, is continuous, g(1)>0 and lim_(x->infty)g(x)=0 It is a fact that for every y between 0 and g(1) the function takes on a value in ( y, g(1) ) How would one show that if: lim_(e->0+)sup{g(x):0<x<e}=0

sengihantq 2022-10-08 Answered
The function g : ( 0 , ) R , is continuous, g ( 1 ) > 0 and
lim x g ( x ) = 0
It is a fact that for every y between 0 and g ( 1 ) the function takes on a value in (   y ,   g ( 1 )   )
How would one show that if:
lim e 0 + sup { g ( x ) : 0 < x < e } = 0
then g attains a maximum value on 0 to infinity. Does g necessarily have to be bounded below for this to hold? Why?
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Answers (1)

ordonansexa
Answered 2022-10-09 Author has 7 answers
Hints
1. Since lim x g ( x ) = 0, there exists R R + such that for x > R, g ( x ) < g ( 1 ).
2. Since lim ϵ 0 + sup { g ( x ) : 0 < x < ϵ } = 0, there exists r R + such that for x ( 0 , r ), g ( x ) < g ( 1 ).
3. r < 1 < R.
4. The set [ r , R ] is a compact set, and a continuous function on a compact domain achieves an absolute maximum on that domain. This is the Extreme Value Theorem.
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