Proving Uni.Conv. of Fn to F on (0,1): Questions & Answers

  • Context: MHB 
  • Thread starter Thread starter bw0young0math
  • Start date Start date
  • Tags Tags
    Function Sequence
Click For Summary
SUMMARY

The discussion centers on the proof of uniform continuity of a function \( f \) on the interval \( (0,1) \) given that a sequence of functions \( f_n \) is uniformly convergent to \( f \) on the same interval. The user proposes a method involving the continuous extension theorem, asserting that if \( f_n \) is uniformly continuous on \( (0,1) \), then \( F_n \) is continuous on the closed interval \([0,1]\). The conclusion drawn is that if \( F_n \) converges uniformly to \( F \) on \([0,1]\), then \( f \) is uniformly continuous on \( (0,1) \).

PREREQUISITES
  • Understanding of uniform continuity and uniform convergence
  • Familiarity with the continuous extension theorem
  • Knowledge of the Triangle Inequality in analysis
  • Basic proficiency in using the \( \varepsilon/3 \) argument in proofs
NEXT STEPS
  • Study the properties of uniform continuity in detail
  • Learn about the continuous extension theorem and its applications
  • Explore the Triangle Inequality and its role in mathematical proofs
  • Practice using \( \varepsilon \)-\( \delta \) definitions in convergence proofs
USEFUL FOR

Mathematics students, particularly those studying real analysis, educators teaching uniform convergence concepts, and researchers exploring function continuity properties.

bw0young0math
Messages
27
Reaction score
0
Hello, today I have a question.

If uni.cont. function sequence fn on (0,1) is uni.conv. to f on (0,1), then f is uni.cont. on (0,1).

The above is true.
The wonder I have is...May I prove the above by this way?

This way: fn is uniform continuous on (0,1). So Fn is continuous on [0,1] (by continuous extention theorem.)
If I can prove Fn is uni. convergent to F on [0,1], (since Fn is cont. on [0,1]) F is cont. on [0,1] and f is uni. cont. on (0,1).

is this way true?
IF it is true, how can I prove Fn is uni.conv. to F on [0,1]?
IF not, could you show me some counter example?Please...(Sadface)(Sadface)(Sadface)
 
Physics news on Phys.org
I think you can use a standard $\varepsilon/3$ argument to prove this. Write out what uniform continuity and uniform convergence mean, using $\epsilon/3$ as the RHS. Then use the Triangle Inequality to finish.
 

Similar threads

Undergrad Create a surjective function from [0,1]^n→S^n
  • · Replies 8 ·
Replies
8
Views
3K
MHB Prove that it converges uniformly
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Question about lifting of a function
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Why does the characteristic function not converge in the L1 space?
  • · Replies 11 ·
Replies
11
Views
5K
MHB Using the Definition: Showing $u \in C^0 [0,1]$
  • · Replies 2 ·
Replies
2
Views
2K
MHB F convex iff Hessian matrix positive semidefinite
  • · Replies 24 ·
Replies
24
Views
4K
MHB Proving Convexity of $f: \mathbb{R} \to \mathbb{R}$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Can a Continuous Curve Only Intersect a Differentiable Curve at the Origin?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Showing Completeness of Metric Spaces: Examples
  • · Replies 1 ·
Replies
1
Views
2K
MHB What is a Semigroup and How Does it Relate to Immeasurable Sets?
  • · Replies 13 ·
Replies
13
Views
4K