May 26, 2016, 12:00-6:00p.m., Avery Hall 347

• Work 5 questions out 6. • Each problem is worth 20 points.

• Parts of each problem don’t necessarily carry the same weight, and they can be unrelated.
• Do not interpret a problem in such a way that it becomes trivial.
• Write on one side of the paper only and hand your work in order.

(1) Consider the function: $f ( x ) = { \frac { x } { 1 - x ^ { 2 } } } , x \in ( 0 , 1 )$

(a) By using the –δ definition of the limit only, prove that f is continuous on (0, 1). (Note: You are not allowed to trivialize the problem by using properties of limits).

(b) Is f uniformly continuous on (0, 1)? Justify your answer.

(2) Let $\{ a _ { k } \} _ { k = 1 } ^ { \infty }$ be a bounded sequence of real numbers and E be given by:

$E : = \left\{ s \in \mathbb { R } : \right.$ : the set $\{ k \in \mathbb { N } : \ a _ { k } \geq s \}$ has at most finitely many elementso.

Prove that lim sup $a _ { k } =$ inf E. k→∞

(3) Assume $( X , d )$ is a compact metric space.

(a) Prove that X is both complete and separable.

(b) Suppose $\{ x _ { k } \} _ { k = 1 } ^ { \infty } \subset X$ is a sequence such that the series $\scriptstyle \sum _ { k = 1 } ^ { \infty } d ( x _ { k } , x _ { k + 1 } )$ is convergent.\
Prove that the sequence $\{ x _ { k } \} _ { k = 1 } ^ { \infty }$ converges in X.

(4) Suppose that $f : [ 0 , 2 ] \to \mathbb { R }$ is continuous on [0, 2], differentiable on (0, 2), and such that $f ( 0 ) = f ( 2 ) = 0 , f ( c ) = 1$ , for some $c \in ( 0 , 2 )$ . Prove that there is an $x \in ( 0 , 2 )$ such that $| f ^ { \prime } ( x ) | > 1$

(5) Let $f _ { n } ( x ) = n ^ { \beta } x \left( 1 - x ^ { 2 } \right) ^ { n } , x \in [ 0 , 1 ] , n \in \mathbb { N } .$

(a) Prove that $\{ f _ { n } \} _ { n = 1 } ^ { \infty }$ converges point-wise on [0, 1] for every $\beta \in \mathbb { R }$ (b) Show that the convergence in part (a) is uniform on [0, 1] for all $\beta < \textstyle { \frac { 1 } { 2 } }$ , but not uniform for any $\beta \geq \textstyle { \frac { 1 } { 2 } }$

(6) (a) Suppose $f : [ - 1 , 1 ] \longrightarrow \mathbb { R }$ is a bounded function that is continuous at 0. Let $\alpha ( x ) = - 1$ for $x \in [ - 1 , 0 ]$ and $\alpha ( x ) = 1$ for $x \in ( 0 , 1 ]$ . Prove that $f \in \mathcal { R } ( \alpha ) [ - 1 , 1 ]$ , i.e., f is Riemann integrable with respect to α on $[ - 1 , 1 ]$ , and $\textstyle \int _ { - 1 } ^ { 1 } f d \alpha = { \dot { 2 } } f ( 0 )$

(b) Let $g : [ 0 , 1 ] \longrightarrow \mathbb { R }$ be a continuous function such that $\begin{array} { r } { \int _ { 0 } ^ { 1 } g ( x ) x ^ { 3 k + 2 } d x = 0 } \end{array}$ , for all $k = 0 , 1 , 2 , \cdots$ . Prove that $g ( x ) = 0$ for all $x \in [ 0 , 1 ]$