Having a little difficulty trying to understand what this question is trying to ask me.
The questions says: "Suppose that f is continuous on [a,b] and f takes only rational values. What can be concluded about f?"
I know that the question is relating to the intermediate value theorem, as it is in that section of the text book, however I'm not sure what can be concluded from this.
My thinking so far has been this:
- f is on a closed, bounded interval, which implies a whole load of things such as continuity etc.
- I'm not sure if the intermediate value theorem works if the function only takes rational numbers, as the reals are dense with irrationals. But they are also dense with the rationals so maybe not?
I don't know how to go about the solution as I'm sure than it is more than "there exists a value c such that a<c<b and f(c) is rational"
Any idea what the question might be asking for???
Thanks in advance,