continuity

[ferhaouim3]

could you help me?
1- is it true that if you strech a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? give reason for your answer.
2- let f be defined on an interval (a,b) and suppose that f(c)#0 at some c where f is continuous. show that there is an interval (c-&, c+&) about c where f has the same sign as f(c).

 

[Guest]

1- WOuldn't it be true because the middle should be in the same position because the left and right are being moved the same??? well, and if it snaps back...

 

[pka]

This is a classic problem.

Lets assume that f(c)<0.
If ε=−f(c)/2 >0 then |x−c|<δ → |f(x)−f(c)|< −f(c)/2.
From that we get 3f(c)/2 < f(x) < f(c)/2.
Or for all x, |x−c|<δ, f(x)<0.