CRN 12109: HW 7

Problem 31. Exercise 4.3.1

Problem 32. Exercise 4.4.7

Problem 33. Exercise 4.4.9. Hint for (b): Consider $f(x)=\sqrt{x}$ on $[0,1]$.

Problem 34. Exercise 4.5.7

Problem 35. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by \[f(x)=\begin{cases}x^2&\mbox{ if }x \mbox{ is rational}\\x^3 & \mbox{ if } x \mbox{ is irrational}\end{cases}\]

1. For which values of $x$ is $f$ continuous? Support your conjecture by a proof.
2. For which values of $x$ is $f$ differentiable? Support your conjecture by a proof.