• Support PF! Buy your school textbooks, materials and every day products Here!

Showing that a function is continuous

  • Thread starter armolinasf
  • Start date
  • #1
196
0

Homework Statement


How would I demonstrate that a function is continuous? Would I just show that it's derivative exists? Thanks for the help.
 

Answers and Replies

  • #2
33,483
5,173

Homework Statement


How would I demonstrate that a function is continuous? Would I just show that it's derivative exists? Thanks for the help.
But what if the function is not differentiable? There are functions that are not differentiable at a given point that happen to be continuous there. For example, y = |x| at x = 0.
 
  • #3
196
0
Then it would be where it's undefined? Also, would I have to examine its end behavior?
 
  • #4
33,483
5,173
No. My example function, f(x) = |x|, is defined everywhere.

How is continuity defined in your book?
 
  • #5
196
0
It's defined by saying that a function is continuous if its graph is an unbroken curve over an interval
 
  • #6
33,483
5,173
The way continuity at a point is usually defined is as a limit:
A function f is continuous at a number a in its domain iff
[tex]\lim_{x \to a}f(x) = f(a)[/tex]
 

Related Threads on Showing that a function is continuous

Replies
2
Views
486
Replies
3
Views
981
Replies
7
Views
930
Replies
39
Views
2K
Replies
3
Views
15K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
5
Views
1K
Replies
1
Views
4K
Top