what are functions?
Here's what Wikipedia says: http://en.wikipedia.org/wiki/Function_(mathematics)
Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input.
If not, then it is a multivalued-function. Multivalued function - Wikipedia, the free encyclopedia
My definition corresponds to, as wikipedia calls it, a well-defined function then.
Sorry, but, no, it doesn't. Yours corresponds to "invertible function". The function, for example, , is a "well defined function", with one and only one output to a particular input but is NOT "one to one and onto" because it does not asociate one and only one input with a particular output.