Midterm 2 Review Sheet

Sequences

A sequence converges to , or if

• for all , there is such that
• the sequence is Cauchy, i.e. for all there is such that

A bounded monotone sequence is convergent

Topology DLC+

A set is compact if

• every sequence in has a convergent subsequence with a limit contained in
• is a subset of , and is also closed and bounded
• every open cover of has a finite subcover

A set is perfect if it is closed and has no isolated points. These sets are always uncountable.

Two sets are separated if and are both empty. A set is disconnected if can be written as a union of two separated sets. A set that is not disconnected is connected

A set is connected if and only if for all with , then .

Examples

Dirichlet’s nowhere continuous function,

Thomae’s function

Altered Sine function

Functional Limits

Functional Limit For , implies that

Sequential Criterion For a function and a limit point , is equivalent to the following statement: for all where and , .

Functional Algebraic Limit Theorem For functions defined on , let and .

1. for all
4. with

Divergence Criterion The limit does not exist if there are two sequences and such that and

A function is monotone increasing if or decreasing for all

Continuity

A function is continuous at if

The function is continuous on if it is continuous at every point.

Discontinuity Criterion If there is a sequence where does not converge to , then is not continuous at .

Algebraic Continuity Theorem Let and be continuous at a point .

1. is continuous at c for all
2. is continuous at
3. is continuous at
4. is continuous at , given the quotient is defined

Composition of continuous functions for and , assume so that is defined. If is continuous at and continuous at , then is continuous at .

Preservation of Compactness for a continuous function , if is compact, then is also compact.

Extreme Value Theorem let be continuous on a compact set . Then there exists such that for all

Uniform Continuity

A function is uniformly continuous if we have

The sequential criterion for nonuniform continuity is similar to previous criterion.

A function that is continuous on a compact set is uniformly continuous on

Intermediate Value Theorem

oops

i don’t remember if this is actually going to be on the midterm, but still worth studying.

Intermediate Value Theorem For continuous, if satisfies or , there is such that .

Preservation of Connected Sets for continuous, if is continuous, then is connected.