Is the Continuity of f(x) and g(x) at x=0 and x=1/2?

  • Thread starter Thread starter sitia
  • Start date Start date
  • Tags Tags
    Function
Click For Summary
SUMMARY

The functions f(x) and g(x) are analyzed for continuity at x=0 and x=1/2. The function f(x) is defined as f(x) = x for x = 1/n (where n is a nonzero integer) and f(x) = 1 - x otherwise. At x=0, f(0) is determined to be 1, and the limit as δ approaches 0 does not converge to f(0), indicating that f(x) is not continuous at this point. The function g(x) is continuous at both x=0 and x=1/2, as it is defined piecewise based on the rationality of x.

PREREQUISITES
  • Understanding of piecewise functions
  • Knowledge of limits and continuity in calculus
  • Familiarity with rational and irrational numbers
  • Basic skills in evaluating functions at specific points
NEXT STEPS
  • Study the definition of continuity in calculus
  • Learn about limits and their properties
  • Explore the behavior of piecewise functions
  • Examine examples of continuity for rational and irrational functions
USEFUL FOR

Students studying calculus, particularly those focusing on continuity and limits, as well as educators seeking to clarify concepts related to piecewise functions.

sitia
Messages
25
Reaction score
0

Homework Statement



I have the function f(x) = x if x= 1/n and n is an integers / {0)
= 1- x otherwise

and the function g(x) = x if x is rational
=1- x if x is irrational

The problems is to find if the above functions are continuous at x=0 and x=1/2 each.
I am really struggling and am not sure how to go about this. Can someone help me figure it out. Please?
Thanks!


Homework Equations





The Attempt at a Solution



For x=0, I believe f(x) is not coninuous as n is an integer and so x cannot equal 0 for x=1/n
However I know that is the worng way of seeing things and I just don't know the steps to take to figuring something like this out.
 
Physics news on Phys.org
f is defined as
f(x) = x if x= 1/n for some nonzero integer n
= 1- x otherwise (i.e. for all other x)
So first question is, what is f(0)?
Next question: does f(δ) get ever closer to that value as δ approaches 0?
 

Similar threads

Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1
  • · Replies 20 ·
Replies
20
Views
3K
Is the given function continuous at x= 0? f(x) = {sin(1/x) if x≠0, 1
  • · Replies 4 ·
Replies
4
Views
4K
f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε
  • · Replies 7 ·
Replies
7
Views
2K
Proof given ##x < y < z## and a twice differentiable function
  • · Replies 8 ·
Replies
8
Views
2K
Determine Convergence/Divergence of Sequence: f(x)=ln(x)^2/x
  • · Replies 34 ·
2
Replies
34
Views
4K
Can this function still be a constant function?
  • · Replies 11 ·
Replies
11
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
  • · Replies 6 ·
Replies
6
Views
2K
Unit normed linear functional on a space of sequences
  • · Replies 13 ·
Replies
13
Views
2K
Prove that Lim x->c f(x)g(x)=0 if Lim x->c f(x)=0 and |g(x)|<K
  • · Replies 2 ·
Replies
2
Views
2K