This midterm will have 20% on series and 80% on metric space topology and continuous functions.

For series, you need to know the basic definitions of convergence, absolute convergence. And various examples that illustrates the differences between these notions. Root test and ratio test for absolute convergence. Read Ross for a good account of examples.

For metric space and topological spaces, pay attention to the notion of open subset, closed subset, compact set. One should know what does these notion means in $\R$, $\R^2$. Given a subset $S \subset X$ of a metric space, one should know what is the induced metric, the induced topology. The relations between compact and closed and bounded set: compact implies closed and bounded for any metric space; for the metric space $\R^n$ (not for any other arbitrary metric space), closed and bounded implies compact.

For continuous functions between metric spaces (and topological spaces), one should know the basic definitions (using $\epsilon-\delta$ language, using open sets, using sequences), and why these various definitions are equivalent. Continuous function preserves compact set and preserves connected set. We also discussed discontinuities on $\R$, and monotone functions. Consider the various examples, e.g $\sin(1/x)$, $x \sin(1/x)$. The monotone function $\sum_n c_n \Theta(x-x_n)$, where $c_n > 0, \sum_n c_n < \infty$ describes the jump sizes, and $E = \{x_n\} \subset \R$ describes the jump locations, and $\Theta(x)$ is the step function: $\Theta(x)=1$ for $x\geq 0$, $\Theta(x)=0$ for $x < 0$.

As a procedure, read textbook Ross and Rudin, do the examples in the textbook. Do the exercises in Ross. Review the HW. Do the exercises in Rudin (harder). Try to form a study group, and ask each other questions.