Thread: 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

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.
We already talked about closed getting mapped to closed.
You can answer this now.

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!