Do functions have to be defined at all values of x?

Question

I'm observing a function that doesnt and undefined y value at x=0.

Is it still classified as a function?

Answer 1

no

th definition is that a function for every x has just one y

an example of a non function y=1/x or  is y =√ ( x -3)

they have "holes" or asymptotes or nothing for y given x.

try 0 in 1/x doesnt work

try -4 in the √x-3 wont work.

Answer 2

You can have holes in the domain.  Happens all the time.  Like, for instance, f(x) = 1/x is undefined at x = 0.  All you have to have for a function is an unambiguous relationship.  One value in the domain leads to only one value in the range.  That value can certainly be "undefined".