[Solution Library] Prove or give a counterexample for each statement. If f is continuous on D and k ∈ R, then $k f$ is continuous on D. If f and f+g are

Question: Prove or give a counterexample for each statement.

If $f$ is continuous on $D$ and $k \in \mathbb{R}$, then $k f$ is continuous on $D$.
If $f$ and $f+g$ are continuous on $D$, then $g$ is continuous on $D$.
If $f$ and $fg$ are continuous on $D$, then $g$ is continuous on $D$.
If $f^{2}$ is continuous on $D$, then $f$ is continuous on $D$.
If $f$ is continuous on $D$, then $f(D)$ is a bounded set.
If $f$ and $g$ are not continuous on $D$, then $f+g$ is not continuous on $D$.
If $f$ and $g$ are not continuous on $D$, then $fg$ is not continuous on $D$.
If $f: D \rightarrow E$ and $g: E \rightarrow F$ are not continuous on $D$ and $E$, respectively, then $g \circ f: D \rightarrow F$ is not continuous on $D$

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[Solution Library] Prove or give a counterexample for each statement. If f is continuous on D and k ∈ R, then $k f$ is continuous on D. If f and f+g are

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