Exploring Non-Continuous Functions in an Interval

In summary: So the set of discontinuous functions cannot be a linear space.In summary, discontinuous functions do not form a linear space because they are not closed under addition and multiplication, as shown by the counterexamples given. This means that the space of discontinuous functions does not satisfy the necessary conditions to be a linear space.
  • #1
batballbat
127
0
why arent non continuous functions in an interval a linear space?
 
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  • #2
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?
 
  • #3
why does f+g have to be continuous? I can see that even for discontinuous functions they are closed under addition and multiplication
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 
  • #7
If it were linear, it would mean that f + g would be discontinuous if f and g are. i don't understand this part
 
  • #8
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.
 
  • #9
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.
 

1. What is a non-continuous function?

A non-continuous function is a mathematical function that has at least one point where it is not defined or is discontinuous, meaning there is a break or jump in the graph of the function.

2. How do you determine if a function is non-continuous?

A function is considered non-continuous if it has any of the following characteristics:
- It has a point of discontinuity, where the function is not defined.
- It has a jump discontinuity, where the function has a sudden change in value.
- It has a removable discontinuity, where the function has a hole or gap in the graph.
- It has an infinite discontinuity, where the function approaches positive or negative infinity at a certain point.

3. What is the significance of exploring non-continuous functions in an interval?

Exploring non-continuous functions in an interval helps us understand the behavior of a function and how it changes within a specific range of values. It also allows us to identify and analyze any discontinuities in the function, which can provide insight into its properties and applications.

4. How are non-continuous functions different from continuous functions?

Non-continuous functions differ from continuous functions in that they have points where they are not defined or have a break/jump in their graph. Continuous functions, on the other hand, are defined for all values within a given interval and have a smooth, unbroken graph.

5. How can we graph non-continuous functions?

To graph non-continuous functions, we first identify the points of discontinuity and mark them on the graph. Then, we graph the function as normal, making sure to leave a gap or break at the points of discontinuity. Additionally, we can use different graphing techniques, such as piecewise functions, to accurately represent the function's behavior and discontinuities.

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