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(See Solution) Let f: D \rightarrow R be continuous. For each of the following, prove or give a counterexample. If D is open, then f(D) is open. If D is closed,

Question: Let $f: D \rightarrow \mathbb{R}$ be continuous. For each of the following, prove or give a counterexample.

If $D$ is open, then $f(D)$ is open.
If $D$ is closed, then $f(D)$ is closed.
If $D$ is not open, then $f(D)$ is not open.
If $D$ is not closed, then $f(D)$ is not closed.
If $D$ is not compact, then $f(D)$ is not compact.
If $D$ is unbounded, then $f(D)$ is unbounded.
If $D$ is finite, then $f(D)$ is finite.
If $D$ is infinite, then $f(D)$ is infinite.
If $D$ is an interval, then $f(D)$ is an interval.
If $D$ is an interval that is not open, then $f(D)$ is an interval that is not open.

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(See Solution) Let f: D \rightarrow R be continuous. For each of the following, prove or give a counterexample. If D is open, then f(D) is open. If D is closed,

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