Existence of a function vs being well-defined?

[merry]

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!

 

[fresh_42]

No, the word function implies well definition, otherwise we have a relation, not a function.

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.

 

[WWGD]

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).