Week 1

Question 1

Answer 1

Question 2

Answer 2

Question 3

Answer 3

And all jump

Question 4

All discontinuities are?

Answer 4

Question 5

Criteria for continuity of a function at a point given by limits of sequences

Answer 5

Question 6

Definition From functions and sequences of convergence

Answer 6

Question 7

Answer 7

Question 8

Answer 8

Question 9

Answer 9

If atleast one of the left or right limit is Infinite

Question 10

Answer 10

However you define f(x_0) , f will not be continuous

Question 11

Answer 11

Question 12

f is continuous at x_0 if? Definition from limits

Answer 12

Question 13

Answer 13

Question 14

Answer 14

Question 15

Of a function at a point

Answer 15

Question 16

Answer 16

Question 17

Answer 17

Question 18

As x->0

Answer 18

Question 19

A set A € R is said to be bounded above if

Answer 19

For A€ (-inf, M] for some M€R

Question 20

A set A € R is said to be bounded below if

Answer 20

A€ [m, infinity) for some m€R

Question 21

If x is real, Neighbourhood of x is

Answer 21

Any open interval (a,b) such that x€(a,b)

Question 22

ε neighbourhood of x?

Answer 22

For ε>0 , (x-ε, x+ε)

Question 23

x is called an interior point of A if

Answer 23

A contains a neighbourhood of x

Question 24

A set B is closed if

Answer 24

It’s complement is open

Question 25

Axiom of completeness

Answer 25

Every set that is bounded above has a supremum, every set that is bounded below has an infimum

Question 26

Define supremum? How to prove that M is supremum

Answer 26

Least upper bound
Prove that it’s an upper bound
Prove there is no M’ that is an upper bound but less than M

Question 27

Define infimum? How to prove that m is infimum?

Answer 27

Greatest lower bound
Prove that it’s a lower bound
Prove there is no m’ that is a lower bound but greater than m

Question 28

Definition of a countable set A on Real line

Answer 28

If it is either finite or there exists a bijection between A and N

Question 29

Which sets are countable, which aren’t

Answer 29

Z , Q are countable
R isnt

Question 30

A sequence is said to be a Cauchy sequence if?

Answer 30

Question 31

Cauchy’s criterion

Answer 31

A sequence converges if and only if it’s a Cauchy sequence

Question 32

Bolzano Weierstrass

Answer 32

Question 33

Define all of the below

Answer 33

The set Δ is called the DOMAIN OF DEFINITION of the function

The set of all of the values when x runs over Δ , f(x) is said to be the RANGE of f, denoted ran(f)

Question 34

Reimann Zeta function

Answer 34

Which converges for s>1

Question 35

Natural domain, define

Answer 35

If f(x) is given by an explicit formula, natural domain is set of values x€R such that the formula makes sense

Question 36

Natural domain of below?

Answer 36

Question 37

Define
A periodic function
An odd function
An even function

Answer 37

Question 38

A function is said to be bounded if ?
Unbounded if?

Answer 38

Question 39

Define punctured neighbourhood

Answer 39

That 0 should be x_0

Question 40

f(x) converges to y_0 as x->x_0 if

Answer 40

Question 41

Write convergence in terms of neighbourhoods

Answer 41

Question 42

For f defined on (a,infinity) a€R, limit of f(x) as x-> infinity is given?

Answer 42

Question 43

For f defined on (a,infinity) a€R, limit of f(x) as x-> infinity is given?
(In terms of neighbourhoods)

Answer 43

Question 44

Dirichlet function

Answer 44

Question 45

Characteristic function of set A

Answer 45