Show that |c|<1|c|<1 ?

Given ff continuous , f:RR->RR
If f(x)!=0 for xin(-oo,c)uu(c,+oo) and

f(1)f(-1)<0

then show that |c|<1

1 Answer
Jan 18, 2018

Solved.

f is continuous in RR and so [-1,1]subeRR .

  • f(1)f(-1)<0

According to Bolzano Theorem (generalization)

EE x_0in(-1,1): f(x_0)=0

Supposed |c|>=1 <=> c>=1 or c<=-1

  • If c>=1 then f(x)!=0 if xin(-oo,c)uu(c,+oo)

However, f(x_0)=0 with x_0in(-1,1) => -1 < x_0 <1<=c => x_0in(-oo,c)

CONTRADICTION!

  • If c<=-1 then f(x)!=0 if xin(-oo,c)uu(c,+oo)

However, f(x_0)=0 with x_0in(-1,1) =>

c<=-1 < x_0<1 => x_0in(c,+oo)

CONTRADICTION!

Therefore,

|c|<1