Does this count as a function?

[Guest]

f(x)=1/(x-x)

Premise: functions can be undefined at certain points and still be functions.
Can a function be discontinuous at every point and still be a function?

If so, can this "function" be put in a piece wise function to make a chunk that is entirely discontinuous.


If these are all false is there a function which has a chunk that is entirely discontinuous?

Thank you.

[HallsofIvy]

The obvious place to start would be with the definition of "function": "A special relationship where each input has a single output."

So for example, f(x)= 0 if x is a rational number and f(x)= 1 if x is an irrational number is a "function" that is not continuous at any value of x.

However, "1/(x- x)" is NOT a function because x- x= 0 for all x and 1/0 is NOT a numerical value.