First consider an increasing function from to which takes all three

values . This is charaterised by two natural numbers and ,

where the function changes from to , and from to . The set of ordered

pairs of naturals is countable (you should already know this, so the set of

increasing functions from to which take all three values is

countable.

Now the set of all functions that we seek is the union of those that take all

three values as considered above, those that take two of the possible values.

You need to show that this latter set of functions is also countable. Then use

the result that the union of two countable sets is also countable.

RonL