Combinations of Continuous Functions

(1) Let $\displaystyle g(x) = \sqrt[3]{x}$.

(a) Prove that $\displaystyle g$ is continuous at $\displaystyle c=0$.

(b) Prove that $\displaystyle g$ is continuous at a point $\displaystyle c \not=0$. (The identity $\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2$) will be helpful)

(2) Using the $\displaystyle \epsilon-\delta$ characterization of continuity (and thus using no previous results about sequences), show that the linear function $\displaystyle f(x)= ax+b$ is continuous at every point of $\displaystyle \mathbb{R}$.

(3)

(a) Using the definition: ( A function $\displaystyle f: A \rightarrow \mathbb{R}$ is continuous at a point $\displaystyle c \in A$ if, for all $\displaystyle \epsilon > 0$, there exists a $\displaystyle \delta > 0$ such that whenever $\displaystyle |x-c| < \delta$ (and $\displaystyle x \in A$) it follows that $\displaystyle |f(x)-f(c)| < \epsilon$. If $\displaystyle f$ is continuous at every point in the domain $\displaystyle A$, then we say that $\displaystyle f$ is continuous on $\displaystyle A$.) show that any function $\displaystyle f$ with domain $\displaystyle \mathbb{Z}$ will necessarily be continuous at every point in its domain.

(b) Show in general that if $\displaystyle c$ is an isolated point of $\displaystyle A$ $\displaystyle \subseteq \mathbb{R}$, then $\displaystyle f: A\rightarrow R$ is continuous at $\displaystyle c$.

(4) Assume $\displaystyle h: \mathbb{R} \rightarrow \mathbb{R} $ is continuous on $\displaystyle \mathbb{R}$ and let $\displaystyle K = {x: h(x) = 0}$. Show that $\displaystyle K$ is a closed set.

If anyone could figure out any of these, I would greatly appreciate it! Thanks!