Week 1 Flashcards
1
Q
A
2
Q
A
3
Q
A
And all jump
4
Q
All discontinuities are?
A
5
Q
Criteria for continuity of a function at a point given by limits of sequences
A
6
Q
Definition From functions and sequences of convergence
A
7
Q
A
8
Q
A
9
Q
A
If atleast one of the left or right limit is Infinite
10
Q
A
However you define f(x_0) , f will not be continuous
11
Q
A
12
Q
f is continuous at x_0 if? Definition from limits
A
13
Q
A
14
Q
A
15
Q
Of a function at a point
A
16
Q
A
17
Q
A
18
Q
As x->0
A
19
Q
A set A € R is said to be bounded above if
A
For A€ (-inf, M] for some M€R
20
Q
A set A € R is said to be bounded below if
A
A€ [m, infinity) for some m€R
21
Q
If x is real, Neighbourhood of x is
A
Any open interval (a,b) such that x€(a,b)
22
Q
ε neighbourhood of x?
A
For ε>0 , (x-ε, x+ε)
23
Q
x is called an interior point of A if
A
A contains a neighbourhood of x
24
Q
A set B is closed if
A
It’s complement is open
25
Axiom of completeness
Every set that is bounded above has a supremum, every set that is bounded below has an infimum
26
Define supremum? How to prove that M is supremum
Least upper bound
Prove that it’s an upper bound
Prove there is no M’ that is an upper bound but less than M
27
Define infimum? How to prove that m is infimum?
Greatest lower bound
Prove that it’s a lower bound
Prove there is no m’ that is a lower bound but greater than m
28
Definition of a countable set A on Real line
If it is either finite or there exists a bijection between A and **N**
29
Which sets are countable, which aren’t
**Z** , **Q** are countable
**R** isnt
30
A sequence is said to be a Cauchy sequence if?
31
Cauchy’s criterion
A sequence converges if and only if it’s a Cauchy sequence
32
Bolzano Weierstrass
33
Define all of the below
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)
34
Reimann Zeta function
Which converges for s>1
35
Natural domain, define
If f(x) is given by an explicit formula, natural domain is set of values x€**R** such that the formula makes sense
36
Natural domain of below?
37
Define
A periodic function
An odd function
An even function
38
A function is said to be bounded if ?
Unbounded if?
39
Define punctured neighbourhood
That 0 should be x_0
40
f(x) converges to y_0 as x->x_0 if
41
Write convergence in terms of neighbourhoods
42
For f defined on (a,infinity) a€**R**, limit of f(x) as x-> infinity is given?
43
For f defined on (a,infinity) a€**R**, limit of f(x) as x-> infinity is given?
(In terms of neighbourhoods)
44
Dirichlet function
45
Characteristic function of set A
