Monotonically decreasing function

[daniel_i_l]

Homework Statement

True or false:
1) If f:R->R is a monotonically decreasing function then every discontinues point of f has finate right and left limits which are unequal.
2) I is some finate segment where a<b and a,b in R. If every continues function defined in I has a maximum and a minimum then I is a closed (in other words [a,b]) segment.

Homework Equations

The Attempt at a Solution

1) I think this is true but how can I prove it? Can someone give me a push in the right direction?
2)True: f(x)=x is continues and defined in any segment but it only has a min and max in an closed one. Is that right?

Thanks.

 

[Dick]

2) looks fine. For 1) try thinking of it this way. Let c be the discontinuous point. For the left limit consider the sequence f(c-1/n) for integers n>=1. The sequence is decreasing and bounded below (by f(c)) so it has a limit L. Can you show R is the left hand limit?