Discontinuous and continuous functions

Click For Summary
SUMMARY

The discussion centers on finding a function that is continuous at 0 but discontinuous at every other point on the real number line. A proposed solution is the piecewise function defined as \( f(x) = x \) for rational numbers and \( f(x) = -x \) for irrational numbers. Participants emphasize the importance of understanding the limit and continuity definitions, particularly the \((\varepsilon, \delta)\) criteria. The conversation highlights the challenge of grasping continuity at a single point while maintaining discontinuity elsewhere.

PREREQUISITES
  • Understanding of continuity and discontinuity in functions
  • Familiarity with piecewise functions
  • Knowledge of rational and irrational numbers
  • Comprehension of the \((\varepsilon, \delta)\) definition of limits
NEXT STEPS
  • Study the properties of piecewise functions in calculus
  • Explore the \((\varepsilon, \delta)\) definition of continuity in depth
  • Learn about the behavior of limits for rational and irrational sequences
  • Investigate examples of functions with specific continuity properties
USEFUL FOR

Mathematics students, educators, and anyone studying real analysis or calculus, particularly those interested in the nuances of continuity and discontinuity in functions.

Carla1985
Messages
91
Reaction score
0
I need to find a function that is continuous at 0 but discontinuous at every other point. IV been stuck on this for hours now :( thankyou
 
Physics news on Phys.org
Re: discontinuous and continuous functions

Carla1985 said:
I need to find a function that is continuous at 0 but discontinuous at every other point. IV been stuck on this for hours now :( thankyou

Hey Carla! ;)

How about:
$$f(x) = \left\{\begin{aligned}
x & \text{ if } x \in \mathbb Q \\
-x & \text{ if } x \in \mathbb R \backslash \mathbb Q
\end{aligned}\right.$$
 
Carla1985 said:
I need to find a function that is continuous at 0 but discontinuous at every other point. IV been stuck on this for hours now :( thankyou

Continuous at just one point ! , then how does the limit exist ?
 
Not a clue. I don't get it at all. The exact wording of a question, just in case iv got it wrong is: "give an example of a function defined on R which is continuous at x=0 and discontinuous at every other point of R". I like Serena, thank you :)
 
ZaidAlyafey said:
Continuous at just one point ! , then how does the limit exist ?

Consider the definition of the limit of a function, using $(\varepsilon, \delta)$-definitions.
Combine it with the definition of a continuous function in a point.
 

Similar threads

Undergrad A non-empty intersection of closures of level sets implies discontinuity
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Create a surjective function from [0,1]^n→S^n
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Proving a function f is continuous given A U B = X
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Question about lifting of a function
  • · Replies 2 ·
Replies
2
Views
2K
MHB Find a function so that the composition is continuous
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad Continuity of linear map from subspace of Euclidean space
  • · Replies 2 ·
Replies
2
Views
2K
High School Finite linear combination of continuous functions is continuous?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad About convergence of complex power series on the circle ##|z - z_0|=r##
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad Compact support functions and law of a random variable
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Understanding the definition of continuous functions
  • · Replies 9 ·
Replies
9
Views
3K