Limits And COntinuity Of Functions

Question 1

When is A a punctured neighbourhood of C

Answer 1

A is a punctured neighbourhood of c is if Edelta such that (c-delta) U (c,c+delta) is a subset of A

Question 2

What is the epsilon delta definition of a limit

Answer 2

Epsilon delta definition of a limit is:
For all epsilon >0 Edelta >0 for all x e D
0 < mod(x-c) < delta implies mod(f(x)- L) < epsilon

Question 3

What does inertia lemma state

Answer 3

Inertia lemma states that if limf(x) =L > M as x tends to c, then there is delta >0 so for all x E D, 0 < mod(x-c) < delta implies f(x) > M

Question 4

What is sequential characterisation of a limit

Answer 4

Sequential characterisation of limit is:
For all c in [a,b] given Xn is a subset of [a,b], Xn tends to c implies f(Xn) tends to f(c)
Xn =/ c for all n

Question 5

What does algebra of limits state

Answer 5

Algebra of limits states:
Limx tends to c f(x) + g(x) = Lf + Lg 
“.       “ af(x)=aLf
“.      “ f(x)*g(x)=Lf*Lg
“.     “ f(x)/g(x)=Lf/Lg if denominators don’t =0

Question 6

What is the definition of a 1 sided limit (right)

Answer 6

Definition of a 1 sided limit (right) is:
Lim tends to c+f(x)= L for all epsilon >0 E delta >0 for all x element of D,
0 < x-c < delta implies mod(f(x)-L) < epsilon.

Question 7

What is the definition of 1 sided limit (left)

Answer 7

Definition of 1 sided limit (left) is;:
Limx tends to c- f(x) =L if for all x in the domain, for any epsilon >0 E delta >0 s.t
-delta < x-c < 0 implies mod(f(x)-L) < epsilon

Question 8

When does limf(x)=L as x tends to c exist

Answer 8

limf(x)=L as x tends to c exists when the 1 sided limits from left and right = L

Question 9

What is the definition of continuity

Answer 9

Definition of continuity is:
F is continuous at c E D if for all epsilon >0 Edelta >0 : for all x E D mod(x-c) < delta implies mod(f(x)-f(c)) < epsilon

Question 10

What is sequential characterisation of continuity

Answer 10

Sequential characterisation of continuity is :
Function f is continuous at c (epsilon delta definition)
For all c in [a,b] given Xn in [a,b] Xn tends to c implies f(Xn) tends to f(c)

Question 11

What is algebra of continuous functions

Answer 11

Algebra of continuous functions is:
If have Continuous functions f and g, then following are continuous
F+g
Af
Fg 
G(c) doesn’t =0, f/g (at c)

Question 12

What is composition of continuous functions

Answer 12

Composition of continuous functions is:

F o g is continuous if f and g are continuous

Question 13

What does intermediate value theorem state

Answer 13

Intermediate value theorem states that:

If f is continuous on [a,b], with f(a) < f(b), then for all y between f(a) and f(b), E c such that f(c)=y

Question 14

What happens if there is no punctured neighbourhood at point c of a function

Answer 14

If there is no punctured neighbourhood at point c of a function then function is continuous at that point