Limits And Continuity 2

Question 1

What does the weierstrass extremal theorem state

Answer 1

Weierstrass extremal theorem states that :
If f is a continuous function [a,b] to R, then there exists p,q element of [a,b] f(q) <=f(x) <=f(p) for all x element of [a,b] (then it attains it bounds)

Question 2

When do we say function f attains it’s supremum or infimum on D

Answer 2

We say function attains it’s supremum on D if:
E p element of D such that f(p)=sup f such that f(p)>=f(x)
Say it attains it’s infimum on D if:
E q element of D s.t f(q)=inf f s.t f(q)<=f(x)

Question 3

What is a strictly increasing and decreasing function

Answer 3

Strictly increasing function is one s.t
X1 < x2 implies f(x1) < f(x2)
Strictly decreasing is one s.t
X1 < X2 implies f(x1) > f(x2)

Question 4

What does the Bolzano weierstrass theorem state

Answer 4

Bolzano weierstrass theorem states that:

If Xn is a bounded sequence (E M >= 0 s.t mod(Xn) < M) in R then Xn has a convergent sequence

Question 5

What is the definition of an inverse bijection f : D to E

Answer 5

Definition of an inverse bijection f: D to E is:
For all x in D, f^-1(f(x)) = x
Or for all G in E
F(f^-1(y)) = y