Question 1.

Prove using $\displaystyle \epsilon - \delta$ arguments that

(i) $\displaystyle f(x) = x^4$ is continuous at x=a, where a not equal to 0.

(ii) Deduce (quoting an appropriate theorem) that $\displaystyle g(y) = y^{1/4}$ is continuous on the interval $\displaystyle [0,\infty)$.

Question 2.

Consider the function

$\displaystyle f(x)=\left\{\begin{array}{cc}cos(\pi x), &\mbox{ } x\leq 2\\x-1, &\mbox{ } 2<x<4\\4, &\mbox{ } x\geq 4\end{array}\right.$

(i) At which points is f(x) continuous?

(ii) Are there points at which f is continuous on the left or on the right but not continuous?

(iii) Give the smallest value of c for which the function f is an injection if its domain is restricted to the interval [c,3]

Thanks a lot for your guidance guys!