FINALLLL

Question 1

perfect

Answer 1

A set P ⊆ R is perfect if it is closed and contains no isolated points.

Question 2

connected

Answer 2

A set that is not disconnected is called a connected set

Question 3

disconnected

Answer 3

Two nonempty sets A, B ⊆ R are separated if Abar ∩ B and
A ∩ Bbar are both empty. A set E ⊆ R is disconnected if it can be written as E = A ∪ B, where A and B are nonempty separated sets.

Question 4

totally disconnected

Answer 4

A set E is totally disconnected if, given any two points x, y ∈ E, there exist separated sets A and B with x ∈ A, y ∈ B, and E = A ∪ B.

Question 5

functional limit

Answer 5

Let f : A → R, and let c be a limit point of the domain A.
We say that limx→c f(x) = L provided that, for all e > 0, there exists a δ > 0 such that whenever 0 < |x − c| < δ (and x ∈ A) it follows that |f(x) − L| < e

Question 6

continuity at a point c

Answer 6

A function f : A → R is continuous at a point c ∈ A if, for
all ε > 0, there exists a δ > 0 such that whenever |x − c| < δ (and x ∈ A) it follows that |f(x) − f(c)| < ε.

Question 7

uniform continuity

Answer 7

A function f : A → R is uniformly continuous on A if for
every ε > 0 there exists a δ > 0 such that |x − y| < δ implies |f(x) − f(y)| < ε.

Question 8

derivative

Answer 8

Let g : A → R be a function defined on an interval A. Given c ∈ A, the derivative of g at c is defined by

g’(c) = lim(x→c) g(x)-g(c)/x-c