Images of sets under continuous maps

Prove that if a mapping $f: X \rightarrow Y$ of metric spaces is continuous and $X$ is compact, then $f(X)$ is compact in $Y$.

Prove that under the mapping given by a continuous function $f(x), x \in [a ; b]$, an arbitrary closed set $F \subset \subset [a ; b]$ is carried into a closed set.

Is it necessarily true that, under a continuous mapping of one metric space into another:

Prove the stated assertion. The continuous image of a compact set is compact.

Prove the stated assertion. A one-to-one continuous mapping of a compact set is a homeomorphism.

Prove the stated assertion. The continuous image of a connected set is a connected set.

Prove the stated assertion. If a metric space is the continuous image of a separable metric space, then it is also a separable space.

Let ( $M, d$ ) be a metric space and $K$ a compact subset of $M$. Suppose that a function $f: K \rightarrow \mathbb {R}$ is continuous. Prove that $f(K)$ is a compact subset of $\mathbb {R}$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous. Show that $f$ maps closed sets to closed sets and maps $F_{\sigma }$ sets to $F_{\sigma }$ sets. Does $f$ map measurable sets to measurable sets?

Let $f$ be a continuous mapping of the compact space $X$ onto the Hausdorff space $Y$. Show that any mapping $g$ of $Y$ into $Z$ for which $g \circ f$ is continuous must itself be continuous.

If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that

for every set $E \subset X$. ($\overline{E}$ denotes the closure of $E$). Show, by an example, that $f(E)$ can be a proper subset of $\overline{f(E)}$.