Existence of a function vs being well-defined?

In summary, the term "function" implies well-definedness and a function must be continuous at a point for its derivative to exist at that point. Well-definedness also applies when a function is defined on a set partitioned by an equivalence relation, meaning its value is independent of the choice of representative.
  • #1
merry
44
0
Hello,

So I am confused on whether the statement that "a function f exists at all points in an open subset U of (say) R" , indicates that it is well-defined on all the points in that subset i.e will the function have a real value on all the points in the subset?
Also, can the derivative of a function exist at a point if the function is not well-defined at the point? For example, if the function goes to infinity at a point, is it possible for its derivative to exist at that point?

thanks!
 
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  • #2
No, the word function implies well definition, otherwise we have a relation, not a function.
merry said:
Also, can the derivative of a function exist at a point if the function is not well-defined at the point?
If a function isn't defined at a certain point, then neither is its derivative there.
 
  • #3
Function must actually be continuous at a point x ( necessary but not sufficient) for the derivative to exist at x.Edit: in my experience, well-definedness applies to cases where the function f is defined on a set partitioned by an equivalence relation . We then say f is well-defined iff( def.) its value is independent of choice of representative, i.e. if [a]= then we must have f(a)=f (b).
 
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1. What is the difference between a function and being well-defined?

A function is a mathematical concept that maps inputs to outputs, while being well-defined means that a mathematical object has a clear and unambiguous definition. A function can only be well-defined if it has a unique output for every input.

2. Why is it important for a function to be well-defined?

If a function is not well-defined, it can lead to contradictions and inconsistencies in mathematical arguments. It is essential for a function to be well-defined in order for it to be used effectively in mathematical equations and proofs.

3. Can a function be well-defined but not exist?

Yes, a function can be well-defined but not exist. This means that it has a clear definition, but there may not be any inputs that result in a valid output. For example, the function f(x) = 1/x is well-defined for all real numbers except for x = 0, but it does not exist at x = 0.

4. Is it possible for a function to exist but not be well-defined?

No, a function cannot exist without being well-defined. In order for a function to exist, it must have a unique output for every input, which is a requirement for being well-defined.

5. How can we determine if a function is well-defined?

To determine if a function is well-defined, we must check that each input has a unique output. This can be done by substituting different values for the input and seeing if the resulting output is the same. If there are any inputs that result in multiple outputs, the function is not well-defined.

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