Real analysis: show that a continuous function is defined for irrationals

Click For Summary
SUMMARY

The discussion centers on proving that a continuous function f defined on the interval (a, b) is equal to zero for all x in (a, b) if f(x) = 0 for all rational numbers x in that interval. The proof strategy involves contradiction, leveraging the properties of continuity and the density of rational numbers in the real numbers. Specifically, it utilizes the fact that if f is continuous and a sequence of rational numbers converges to a real number, then the function values must also converge to the function value at that real number.

PREREQUISITES
  • Understanding of continuity in real analysis
  • Knowledge of sequences and limits
  • Familiarity with the density of rational numbers in real numbers
  • Experience with proof techniques, particularly proof by contradiction
NEXT STEPS
  • Study the properties of continuous functions in real analysis
  • Learn about sequences and their convergence in the context of real numbers
  • Explore the concept of density in topology, particularly regarding rational and irrational numbers
  • Practice proof techniques, focusing on proof by contradiction in mathematical analysis
USEFUL FOR

This discussion is beneficial for students and educators in real analysis, particularly those focusing on continuity and the properties of functions. It is also useful for mathematicians interested in foundational proofs involving rational and irrational numbers.

nps12345
Messages
1
Reaction score
0
Let f be a continuous function defined on (a, b). Supposed f(x)=0 for all rational numbers x in (a, b). Prove that f(x)=0 on (a, b).

i don't even know where to start...any tips just to point me in the right direction?
 
Physics news on Phys.org
There's a couple of ways you can approach this problem. Did you try proof by contradiction?

Let f be a continuous function defined on (a, b). Assume f(x)=0 for all rational numbers x in (a, b) and assume that f(x)≠0 on (a, b), so...
 
You'll need two things for this problem:

1) If f is continuous and if [itex]x_n\rightarrow x[/itex], then [itex]f(x_n)\rightarrow f(x)[/itex].

2) For every real number x there exists a sequence of rational numbers that converges to x. This is saying that [itex]\mathbb{Q}[/itex] is dense in [itex]\mathbb{R}[/itex]

Try to do something with these things...
 

Similar threads

Undergrad Proving Continuous Functions in Smooth Infinitesimal Analysis
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad A correct definition of sequential right continuity of a function
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
7K
Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Ambiguity of the term "indefinite integral"
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad On (real) entire functions and the identity theorem
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Open interval or Closed interval in defining convex function
  • · Replies 4 ·
Replies
4
Views
3K