Continuity And Integrability Fundamentaux

Question 1

What is the definition of continuity

Answer 1

Definition of continuity is:
F is continuous at x if for all epsilon E delta > 0 s.t for all y in domain mod(y-x) < delta implies mod(f(y) - f(x)) < epsilon

Question 2

What is definition of continuity (sequences)

Answer 2

Definition of continuity (sequences) is:
F is continuous at x if for any sequence Xk with x = lim k tends to inf Xk we have f(x) = lim k tends to inf f(Xk)

Question 3

When is function f Lipschitz continuous

Answer 3

Function f is Lipschitz continuous if exists L > 0 s.t for all x,y in domain :
Mod(f(y)-f(x)) <= L*mod(y-x)

Question 4

What does the theorem of uniform continuity state

Answer 4

Theorem of uniform continuity states that if I is a closed bounded interval and f : I to R is continuous, then f is uniformly continuous

Question 5

What does the weierstrass extreme value theorem state

Answer 5

Weierstrass extreme value theorem states that if f : I to R is continuous, then f is bounded and attains it’s infimum and supremum. (I is closed bounded interval)

Question 6

When is a function f : [a,b] to R Riemann integrable

Answer 6

A function f : [a,b] to R is Riemann integrable when:
For every epsilon > 0 exists a subdivision delta of [a,b] (a= X0 < X1,…,< Xm =b) such that
U(f,delta) - L(f,delta) = sum n=1 to M (sup In (f) - inf In (f))* mod(In) < epsilon
In = [Xn-1,Xn] , mod(In) = Xn - Xn-1

Question 7

What does fundamental theorem of calculus state

Answer 7

Fundamental theorem of calculus states that:
If F is anti derivative of f in [a,b] and f is Riemann integrable on [a,b] then integral of b down to a f(x) dx = F(b) - F(a)

Question 8

What does the mean value theorem state

Answer 8

Mean value theorem states that: if a < b and f is differentiable on (a,b7 then exists c in (a,b) s.t f’(c) = (f(b) - f(a))/ b-a