Showing that a function is continuous

Click For Summary

Homework Help Overview

The discussion revolves around demonstrating the continuity of a function, exploring definitions and conditions related to continuity and differentiability.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss whether showing a function's derivative exists is sufficient for proving continuity, and question the implications of differentiability on continuity. They also explore the definition of continuity and the role of limits.

Discussion Status

The discussion is actively exploring different definitions and interpretations of continuity. Participants are questioning assumptions about differentiability and its relationship to continuity, while also clarifying the formal definition of continuity at a point.

Contextual Notes

There is mention of specific functions, such as y = |x|, and the need to consider their behavior at certain points, as well as the definition of continuity provided in the original poster's textbook.

armolinasf
Messages
195
Reaction score
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.
 
Physics news on Phys.org
armolinasf said:

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.
 
Then it would be where it's undefined? Also, would I have to examine its end behavior?
 
No. My example function, f(x) = |x|, is defined everywhere.

How is continuity defined in your book?
 
It's defined by saying that a function is continuous if its graph is an unbroken curve over an interval
 
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]
 

Similar threads

Show that the given function is continuous
  • · Replies 27 ·
Replies
27
Views
2K
Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1
  • · Replies 20 ·
Replies
20
Views
3K
Uniform Continuity of functions
  • · Replies 40 ·
2
Replies
40
Views
5K
Can this function still be a constant function?
  • · Replies 11 ·
Replies
11
Views
2K
Study Function: Continuity, Derivatives & Differentiability
  • · Replies 6 ·
Replies
6
Views
2K
f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε
  • · Replies 7 ·
Replies
7
Views
2K
Help Needed Proving Implication for Linear Functional on Banach Space
  • · Replies 2 ·
Replies
2
Views
1K
Is the given function continuous at x= 0? f(x) = {sin(1/x) if x≠0, 1
  • · Replies 4 ·
Replies
4
Views
4K
Assume that if the real-valued function h(x) is Lipschitz continuous...
  • · Replies 22 ·
Replies
22
Views
2K
Why is this valid? Continuity of Functions Problem
  • · Replies 3 ·
Replies
3
Views
1K