This question shows a common problem that many students have with definitions in math. It comes up when you define a word that already has connotations, in this case "function". The only data you can use in proofs based on this new definition are the properties given by the definition (until someone has proved some theorems you can use). If the fact that a function in the general usage of mathematical English is determined by its values is not given in the definition, you can't use it. In general, if you see a math definition of a word, throw away its connotations!

The problems in understanding mathematical English, even in formally written papers, is well described in the book The Language of mathe­matics, by Mohan Ganesalingam, and many other examples are given in my website athttp://www.abstractmath.org/MM/MMLangMath.htm