Question

Show that `f` is not constant and find `f`?

Given `f:RR->QQ` continuous in `RR` with `f(2004)=2013`. Show that `f` is not constant and find `f`

Answer 1

The question should say "Show that `f` is a constant function."

Explanation:

Use the intermediate value theorem.

Suppose that `f` is a function with domain `RR` and `f` is continuous on `RR`.

We shall show that the image of `f` (the range of `f`) includes some irrational numbers.

If `f` is not constant, then there is an `r in RR` with `f(r) =s != 2013`

But now `f` is continuous on the closed interval with endpoints `r` and `2004`, so `f` must attain every value between `s` and `2013`.

There are irrational numbers between `s` and `2013`, so the image of `f` includes some irrational numbers.