Continuity Adv. Calc 1

  • Thread starter chief12
  • Start date
  • #1
11
0

Homework Statement


part 1)Show the function a(x)=|x| is a continuous function from R to R;

part 2)

Prove that if the functions f: D--> R is continuous at x=a, then l f l (absolute value of f) is also continuous at x=a.


Homework Equations





The Attempt at a Solution


part 1) since a(x) = |x|, then given any [tex]\epsilon[/tex] > 0, then for all l f(x) - f(a) l < [tex]\epsilon[/tex], since l x - a l < [tex]\delta[/tex] when delta = epsilon, since f(x) = l x l


part 2) since f is continuous, then the absolute value of f is also continuous since it doesnt change any of the relationships

then given any [tex]\epsilon[/tex] > 0, then for all l f(x) - f(a) l < [tex]\epsilon[/tex], since l x - a l < [tex]\delta[/tex] when delta = epsilon, since f(x) = l x l
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
770

Homework Statement


part 1)Show the function a(x)=|x| is a continuous function from R to R;

part 2)

Prove that if the functions f: D--> R is continuous at x=a, then l f l (absolute value of f) is also continuous at x=a.


Homework Equations





The Attempt at a Solution


part 1) since a(x) = |x|, then given any [tex]\epsilon[/tex] > 0, then for all l f(x) - f(a) l < [tex]\epsilon[/tex], since l x - a l < [tex]\delta[/tex] when delta = epsilon, since f(x) = l x l

Not quite. Both the argument and the writeup need improvement.

Your final proof for part 1 should look something like this:

Suppose ε > 0. Pick δ = ε (if that is what works). Then if |x-a| < δ you have.... At this point you need to give an argument to show that

|f(x) - f(a)| = | |x| - |a| |< ε

I used f(x) instead of a(x) for your function so it doesn't use "a" twice.
 

Related Threads on Continuity Adv. Calc 1

  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
3
Views
885
  • Last Post
Replies
10
Views
1K
  • Last Post
Replies
9
Views
785
  • Last Post
Replies
6
Views
871
  • Last Post
Replies
3
Views
917
Replies
4
Views
3K
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
5
Views
1K
Top