Thread: counterexamples in real analysis

Prove or give a counterexample for each statement
A) If f^2 is continuous on D, then f is continuous on D
B) If f is continuous on D, then f(D) is a bounded subset
C) if f and g are not continuous on D, then f + g is not continuous on D

Originally Posted by luckyc1423

Prove or give a counterexample for each statement
A) If f^2 is continuous on D, then f is continuous on D

Yes.

If f(x)>=0 is continous.
Then, sqrt(f(x)) is continous.

Then,
sqrt(f^2)=|f| is continous.
But if |f| is contonous it must be that f was continous.

B) If f is continuous on D, then f(D) is a bounded subset

False.
Consider,
1/x on (0,1)

C) if f and g are not continuous on D, then f + g is not continuous on D

I have an elegant approach to this one.