chapter 8

Question 1

Definition of a continuous function

Answer 1

Let A C R and f:A->R. f is continuous on A if for all a in A and E > 0 there exists delta > 0 such that if y E A and |a-y|<= delta then |f(a) - f(y)| <= E.

Question 2

definition of a uniformly continuous function

Answer 2

Let A C R and f:A->R. We say f is uniformly continuous on A if for all E > 0 there exists delta > 0 such that if x,y E A and |x-y|<= delta then |f(x) - f(y)| <= E.

Question 3

if f is not uniformly continuous

Answer 3

Let A C R and f:A->R. We say f is NOT uniformly continuous on A if there exists E > 0 such that for all delta > 0 there exists x,y E A and |x-y|<= delta then |f(x) - f(y)| >E.

Question 4

theorem of a continuous function on a closed bounded interval being uniformly continuous

Answer 4

let a<=b for a,b E R and f:[a,b]->R be continous then f is uniformly continuous on [a,b].

Question 5

define a step function

Answer 5

psi n(x)= f(xk) for xE(xk-1, xk) where xk= a+(b-a)k/n.

Question 6

limn->inf sup|f(x)-psi n (x)|=0 xE[a,b] is equivalent to

Answer 6

suppose f:[a,b]-> R is uniformly continuous on A and psi n(x), nEN is defined by psin(x)=f(xk) for all xE(xk-1,xk] then for all E > 0 there exists NEN such that for all n>=N |f(x) - psin(x)| <= E. for all xE[a,b].

Question 7

pointwise convergence definition

Answer 7

let (fn) be a sequence of functions where fn:A->R. We say that (fn) converges pointwise to f:A->R if for all xEA and for all E>0 there exists NEN such that if nEN and n>=N then |fn(x)-f(x)|<= E.
This is equivalent to if for all xEA and n>=N then lim n->inf fn(x)=f(x).

Question 8

definition of uniform convergence

Answer 8

let (fn) be a sequence of functions where fn:A->R. We say (fn) converges uniformly to f:A->R if for all E>0, there exists NEN such that if nEN and n>=N and xEA then |fn(x)-f(x)|<= E.
This is equivalent to for all E>0 there exists NEN such that if n>= N and nEN then supxEA |fn(x)-f(x)| <=E.

Question 9

Weierstrass theorem

Answer 9

let fn:[a,b]-> R be a sequence of functions which converge uniformly to f:[a,b]->R. If the funcitons (fn) are continuous on [a,b] then f is continuous on [a,b].

Question 10

define supremum norm

Answer 10

let A=[a,b] be a closed interval and f:A->R be bounded then the supremum norm of f is defined as ||f||:=sup{|f(x)| :xEA}.

Question 11

another way of saying fn converge uniformly using limits

Answer 11

if (fn) are bounded on[a,b,] and converge uniformly to f:[a,b]->R then lim n->inf ||fn-f||=0.

Question 12

definition of fn being cauchy

Answer 12

we denote C([a,b]) the set of all uniformly continuous function of [a,b]. We say (fn) wehre fn E C([a,b]) for all nEN is a CAUCHY SEQUENCE in C([a,b]) if for all E>0 there exists NEN such that for m,n>N then ||fn-fm||<=E.

Question 13

fn is cauchy definition of fn uniformly convergent limit

Answer 13

let (fn) be a sequence of function in C([a,b]) and fE C([a,b]) such that lim n->inf ||fn-f||=0. Then (fn) is cauchy.