Is every empty set function continuous?

In summary, Rudin's definition of continuity of function does not specify X as a non-empty set, but supposes that p is in X. If applied to an empty set X, a function with a domain E which is a subset of X would be considered continuous at p. In regards to a definition containing contradicting statements, if this were to happen in a theorem, the theorem would not be true.
  • #1
julypraise
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Can I use the definition of continuity of function from Baby Rudin, setting X as empty set?

Rudin does not specify X is a non-empty set but he supposes p is in X.

Anyway if I use it for empty set X, then is a function with a domain E which is a subset of X continuous at p?

One more extra question: If a definition contains two statements that contradict each other in its hypotheses, what happens? (I know if a theorem contains these things, then the theorem becomes vacuously true.)

Note: for the Rudin's definition, please look at the attatched file.
 

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  • #2
The function in question is not continuous at any particular point p, because there aren't any points. However, the functionis continuous on the empty set, because it had no points. Another way to think about it is if the function is discontinuous you have to be able to produce a point where continuity fails, and you can't

I think you're confused by the definition of vacuously true. If two statements in a theorem contradict each other, the theorem can't be true
 

Related to Is every empty set function continuous?

1. What is an empty set function?

An empty set function is a mathematical function that has no input values. It is also known as a nullary function or a constant function, as it always returns the same output value regardless of the input.

2. Is every empty set function continuous?

Yes, every empty set function is continuous. Since an empty set function has no input values, there are no gaps or jumps in its domain. Therefore, it is considered to be continuous at all points.

3. How is continuity defined in mathematics?

Continuity in mathematics refers to the property of a function where small changes in the input value result in small changes in the output value. In other words, a function is continuous if there are no breaks or gaps in its graph.

4. Can an empty set function be discontinuous?

No, an empty set function cannot be discontinuous. Since it has no input values, there is no possibility for it to have a break or gap in its graph. Therefore, it is always considered to be continuous.

5. What is the significance of an empty set function being continuous?

The continuity of an empty set function may seem trivial, but it is essential in certain mathematical concepts, such as topology and measure theory. It also helps in simplifying certain mathematical proofs and calculations.

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