What does a function takes on mean?

[Nikitin]

What does "a function takes on" mean?

Hey! I'm currently learning Picard's theorem, but I don't quite understand it because of difficult words..

In the text, it uses the term "takes on", ie

"If f(z) is analytic and has an isolated essential singularity at a point z_0, it takes on every value, with at most one exceptional value in an arbitrarily small ε-neighbourhood of z_0".

What does "take on" mean? Are they talking about the function's range? And what do they mean with "exceptional value"?

 

[Stephen Tashi]

What does "take on" mean? Are they talking about the function's range?

Yes - or technically the "image" of the function. http://en.wikipedia.org/wiki/Codomain

And what do they mean with "exceptional value"?

It means a complex number c such that y=c is an exception to the statement "the function's image contains the value y". That is, y = c makes the statement false.