Establishing that a function is discontinous

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In summary, the definition of continuity in topology states that a function is continuous if the inverse image of every open set in the codomain is open in the domain. This definition is useful for proving more general cases, but it can be challenging to apply to trivially discontinuous functions, such as the one given in the conversation. By choosing a small open set around the discontinuity, it becomes clear that the inverse image of that set is not open in the domain, proving that the function is not continuous.
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center o bass
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According to the definition of continuity in topology a function f:X -> Y is continuous if for every open set V in Y, the set ##f(V)^{-1}## is open in X. I have found this definition powerfull in dealing with proofs of a more general nature, but when presented with the trivially discontinous function from the reals to the reals

$$f(x) = 1 \ \text{for} \ x< 0, \ 0 \ \text{for} \ \ x \geq 0 $$

I have problems to argue why this function is obviously not continuous from the above definition.
Which open set do I choose in it's codomain which is not open in it's domain?
 
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The set (-1/2,1/2) (or any open set containing zero and not 1) has as its inverse image [itex] [0,\infty) [/itex] which is not open.

Basically as long as you include a small open set around a discontinuity you are good to go if the set is small enough.
 
  • #3
Office_Shredder said:
The set (-1/2,1/2) (or any open set containing zero and not 1) has as its inverse image [itex] [0,\infty) [/itex] which is not open.

Basically as long as you include a small open set around a discontinuity you are good to go if the set is small enough.

Thanks! It's clear now. My confusion lied in trying to construct open sets in the image of f instead of the codomain.
 

Related to Establishing that a function is discontinous

1. What is a discontinuous function?

A discontinuous function is a mathematical function that has a break or gap in its graph. This means that the function is not continuous and cannot be drawn without lifting the pencil from the paper.

2. How do you determine if a function is discontinuous?

To determine if a function is discontinuous, you must analyze its graph and look for any breaks or gaps. You can also check if the function satisfies the three criteria for continuity: the function must be defined at the point in question, the limit of the function at that point must exist, and the limit must equal the value of the function at that point.

3. What are the types of discontinuities?

There are three types of discontinuities: removable, jump, and infinite. A removable discontinuity occurs when a point is missing from the graph, but it can be filled in without creating a break. A jump discontinuity occurs when the graph has a sudden jump or gap. An infinite discontinuity occurs when the function approaches positive or negative infinity at the point in question.

4. How do you prove that a function is discontinuous?

To prove that a function is discontinuous, you must show that it does not satisfy the three criteria for continuity. This can be done by finding a point where the function is not defined, or by showing that the limit of the function does not exist or is not equal to the value of the function at that point.

5. Why is it important to establish that a function is discontinuous?

Establishing that a function is discontinuous is important because it helps us understand the behavior of the function and its graph. Discontinuous functions have different properties and characteristics compared to continuous functions, and this knowledge can be used in various mathematical applications and problem-solving situations.

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