Why Aren't Non-Continuous Functions in an Interval Considered a Linear Space?

  • Context: Graduate 
  • Thread starter Thread starter batballbat
  • Start date Start date
  • Tags Tags
    Functions Interval
Click For Summary

Discussion Overview

The discussion centers around the question of why non-continuous functions defined on an interval are not considered a linear space. Participants explore the properties of discontinuous functions in relation to the axioms of linear spaces, particularly focusing on closure under addition and the existence of an additive identity.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the classification of non-continuous functions as a linear space.
  • Another participant suggests finding a counterexample to demonstrate closure under addition, hinting at the possibility of two discontinuous functions summing to a continuous function.
  • A participant argues that discontinuous functions can be closed under addition and multiplication, challenging the necessity of continuity for the sum of two functions.
  • Examples of discontinuous functions are provided, where two specific functions, f and g, are shown to sum to a continuous function, raising questions about the closure property.
  • Further examples are introduced to illustrate that discontinuous functions do not satisfy closure under multiplication.
  • Clarifications are made regarding the definition of a linear space, emphasizing that if two functions are in the space, their sum must also be in the space.
  • Another participant points out that every linear space must contain an additive identity, which is the zero function, a continuous function not included in the set of discontinuous functions.

Areas of Agreement / Disagreement

Participants express differing views on whether discontinuous functions can form a linear space, with some arguing against it based on closure properties, while others challenge the necessity of continuity in the sum of discontinuous functions. The discussion remains unresolved as multiple competing views are presented.

Contextual Notes

Participants reference specific properties of linear spaces and provide examples to illustrate their points, but there is no consensus on the implications of these examples for the classification of discontinuous functions.

batballbat
Messages
127
Reaction score
0
why arent non continuous functions in an interval a linear space?
 
Physics news on Phys.org
Why don't you try to find a counterexample?

Hint: try looking at closure under addition: can you find two discontinuous functions f and g such that f + g is continuous?
 
why does f+g have to be continuous? I can see that even for discontinuous functions they are closed under addition and multiplication
 
suppose our interval is [0,1]. define f:[0,1]→R by:

f(x) = -1, for 0 ≤ x < 1/2
f(x) = 1, for 1/2 ≤ x ≤ 1.

clearly, f is discontinuous (at 1/2).

now define g:[0,1]→R by:

g(x) = 1 for 0 ≤ x < 1/2
gx) = -1, for 1/2 ≤ x ≤ 1.

again, g(x) is discontinuous (at 1/2).

but (f+g)(x) = 0, for all x in [0,1], and constant functions are continuous.
 
In other words, "discontinuous functions" are NOT closed under addition. For a counter example to closure under multiplication, let f(x)= 1 if x is rational, -1 if x is irrational and let g(x)= -1 if x is rational, 1 if x is irrational.
 
batballbat said:
why does f+g have to be continuous?

Because you wanted to show that the space of discontinuous is not linear. If it were linear, it would mean that f + g would be discontinuous if f and g are.
 
If it were linear, it would mean that f + g would be discontinuous if f and g are. i don't understand this part
 
By definition, a linear space V must satisfy: if x and y are in V, so is x+y.

If V would be the set of discontinuous functions, then the above becomes: if x and y are discontinuous functions, so is x+y.

The above example shows that this is false, hence the discontinuous functions do not form a linear space.
 
thanks
 
  • #10
Or, perhaps much more simply, every linear space must contain an additive identity. Since the addition here is ordinary addition of functions, the "additive identity" is the 0 function (f(x)= 0 for all x). That is a continuous function and so is NOT in the set of discontinuous functions.
 

Similar threads

Insights Why Vector Spaces Explain The World: A Historical Perspective
  • · Replies 0 ·
Replies
0
Views
9K
Undergrad Proving linear independence of two functions in a vector space
  • · Replies 5 ·
Replies
5
Views
2K
Graduate What are Modular Lie Algebras and How Do They Apply to Function Spaces?
  • · Replies 5 ·
Replies
5
Views
3K
MHB Dimension of Dirac Functionals in $V$: Find the Answer
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Wronskian: null space equals the span of independent functions
  • · Replies 23 ·
Replies
23
Views
2K
MHB Is non continuous function also not Bounded ?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Real function inner product space
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Open interval or Closed interval in defining convex function
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Want to understand how to express the derivative as a matrix
  • · Replies 8 ·
Replies
8
Views
3K
MHB Is an increasing monotonic function in a closed interval also continuous?
  • · Replies 1 ·
Replies
1
Views
2K