real analysis

a) False, let $A=\Bbb R$, $B=[0,1]$. Then $\partial(A\times B)=\Bbb R \times \{0,1\}$, but $\partial A=\emptyset$ so $\partial A\times \partial B =\emptyset$. Hence here $\partial(A\times B)\neq \partial A\times \partial B$.

b) This is true, since $|f_n(0)-f(0)| ≤ \sup_{x\in\Bbb R}|f_n(x)-f(x)|$ and the right-hand side of the inequality goes to $0$ with $n\to\infty$.

c) This is true, since $M$ is an upper bound of $A$ if and only if $M ≥ x$ for all $x\in A$. Multiplying both sides by $c$ switches the inequality and you get that
$M$ is an upper bound of $A$ $\iff$ $cM ≤ cx$ for all $x\in A$ $\iff$ $cM$ is a lower bound of $cA$. Hence $\inf(cA)=c\sup(A)$.

d) This is not true. For example the function $f(x)=x+\cos(x)$ has $f'(x)=0$ whenever $x=\pi+2\pi k$ for some $k\in \Bbb N$, but it has no extrema since $f'(x)=1-\sin(x)≥0$ for all $x$.