The following are true or false variances of determining whether a function is continuous.
1. If f(K) is bounded, then f is continuous.
False. A bounded step function such as the characteristic function would serve as a counterexample.
2. If f(K) is sequentially compact whenever K is sequentially, then f is continuous.
False. The characteristic function serves as a counterexample
since it is bounded and closed, but it is not continuous.
(I know the converse is true: if f is continuous and K is sequentially compact, then f(K) is sequentially compact. )
3. If f(A) is closed whenever A is closed, then f is continuous.
False. Consider the function
It is closed on the real line but it is not continuous. Is there a better counterexample for this?
4. If f(A) is open whenever A is open, then f is continuous.
False. I was thinking f(x) = 1/x. The range is , which is open, but f is not continuous.