# Thread: counter examples with real functions

1. ## counter examples with real functions

Let f->R be continuous. For each of the following, prove or give a counter example.

A) if D is open, then f(D) is open
B) if D is closed, then f(D) is closed
D) if D is not closed, then f(D) is not closed
E) if D is not compact, then f(D) is not compact
F) if D is unbounded, then f(D) is unbounded
G) if D is finite, then f(D) is finite
H) if D is infinite, then f(D is infinite
I) if D is an interval, then f(D) is an interval

2. Originally Posted by slowcurv99
Let f: D->R be continuous. For each of the following, prove or give a counter example.
Say D is not empty.
A) if D is open, then f(D) is open
Consider f(x)=1
B) if D is closed, then f(D) is closed
Consider f(x)=1
D) if D is not closed, then f(D) is not closed
Consider f(x)=x^2 on (0,1).
E) if D is not compact, then f(D) is not compact
Heine-Borel Theorem: Compact = Closed + Bounded.
F) if D is unbounded, then f(D) is unbounded
Consider D=(-oo,+oo) and f(x)=0.
G) if D is finite, then f(D) is finite
Consider f(x)=1/x on (0,1)
H) if D is infinite, then f(D) is infinite
How is that different from unbounded?
I) if D is an interval, then f(D) is an interval
It is either an interval or a single point!