Continuous functions and dense sets

Let $X$ and $Y$ be metric spaces, and $f: X \rightarrow Y$ a continuous surjective mapping. Prove that if $M$ is an everywhere dense set in $X$, then its image $f(M)$ is an everywhere dense set in $Y$.

A set $B$ is called dense in a set $A$ if the closure of $B$ contains $A$, i.e., $\bar{B} \supset A$. Let the function $f$ be continuous on $X$, let the set $A$ be dense in $X$, let $f(X)$ be the set of all values of the function $f(x)$ for $x \in X$, and let $f(A)$ be the set of all values of the function for $x \in A$. Prove that $f(A)$ is dense in $f(X)$.

Prove that if $f$ is a continuous mapping of a space $X$ onto a space $Y$ and $E$ is a set dense in $X$, then $f(E)$ is a set dense in $Y$.

Let ( $M, d$ ) be a metric space, let $f: M \rightarrow \mathbb {R}$ be continuous, and let $E \subset M$. Prove that $\sup \left\{ \left|f(x)\right|: x \in E\right\} =\sup \left\{ \left|f(x)\right|: x \in \bar{E}\right\}$.

Let ( $M, d$ ) be a metric space and $A \subset M$. Suppose that functions $f, g: M \rightarrow \mathbb {R}$ are continuous and $f(x)=g(x)$ when $x \in A$. Prove that $f(x)=g(x)$ for all $x \in \bar{A}$.

Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$. If $g(p) = f(p)$ for all $p \in E$, prove that $g(p) = f(p)$ for all $p \in X$. (In other words, a continuous mapping is determined by its values on a dense subset of its domain).