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

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- April 9th 2007, 07:58 AMluckyc1423counterexamples 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 - April 10th 2007, 06:31 PMThePerfectHacker
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.

Quote:

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

Consider,

1/x on (0,1)

Quote:

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