The function g:(0,infty)->R, is continuous, g(1)>0 and lim_(x->infty)g(x)=0 Show that for every y between 0 and g(1) the function takes on a value in ( y, g(1) )

Graham Beasley

Graham Beasley

Answered question

2022-07-20

The function g : ( 0 , ) R , is continuous, g ( 1 ) > 0 and
lim x g ( x ) = 0
Show that for every y between 0 and g ( 1 ) the function takes on a value in (   y ,   g ( 1 )   )

Answer & Explanation

Arthur Gillespie

Arthur Gillespie

Beginner2022-07-21Added 10 answers

Given 0 < y < g ( 1 ), choose x 0 > 1 so that g ( x 0 ) < y (this is always possible since lim x g ( x ) = 0. Then by the IVT, since g ( 1 ) > y > g ( x 0 ), for every u ( y , g ( 1 ) ) there is an x 1 ( 1 , x 0 ) such that g ( x 1 ) = u, as desiblack.

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