Is a Function Continuous in a Neighborhood If It's Continuous at a Point?

  • Context: Undergrad 
  • Thread starter Thread starter JG89
  • Start date Start date
  • Tags Tags
    Continuity
Click For Summary
SUMMARY

The discussion centers on the concept of continuity in mathematical functions, specifically addressing whether a function that is continuous at a point is also continuous in a neighborhood around that point. The example provided is the function f(x) defined as f(x) = x for rational x and f(x) = 0 for irrational x. This function is continuous at x = 0 but not in any neighborhood around it, illustrating that continuity at a point does not imply continuity in the surrounding area.

PREREQUISITES
  • Understanding of mathematical continuity
  • Familiarity with rational and irrational numbers
  • Basic knowledge of function definitions
  • Concept of neighborhoods in topology
NEXT STEPS
  • Study the formal definition of continuity in calculus
  • Explore examples of discontinuous functions
  • Learn about neighborhoods and limits in topology
  • Investigate the implications of continuity in real analysis
USEFUL FOR

Mathematics students, educators, and anyone interested in advanced calculus or real analysis concepts related to function behavior and continuity.

JG89
Messages
724
Reaction score
1
If f is continuous at x = a, then is it continuous in some neighborhood of x = a as well?
 
Physics news on Phys.org
No
A standard example is
f(x)=x x rational
f(x)=0 x irrational

f is continuous at x=0
 
lurflurf said:
No
A standard example is
f(x)=x x rational
f(x)=0 x irrational

f is continuous at x=0

And nowhere else. Not even very close to 0.

--Elucidus
 

Similar threads

Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad A correct definition of sequential right continuity of a function
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Continuity of Mean Value Theorem
  • · Replies 12 ·
Replies
12
Views
3K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Understanding Continuous Functions: Examining f'(7) Undefined
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Proving Continuous Functions in Smooth Infinitesimal Analysis
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Doubt about theorem in Calculus on Manifolds
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Can Schwarz's Theorem be used "Backwards" to prove continuity?
  • · Replies 21 ·
Replies
21
Views
6K
MHB Differentiability and continuity
  • · Replies 1 ·
Replies
1
Views
2K
MHB Is non continuous function also not Bounded ?
  • · Replies 1 ·
Replies
1
Views
1K