0

Let $f(x)$ be a continuous function defined for $1≤x≤3$. If $f(x)$ takes rational values for all $x$ and $f(2)=5$, what is the value of $f(1.5)$?

How do I approach questions like these?

user744725
  • 47
  • 2
    Using the intermediate value theorem. If the function takes another value different from $5$, then it will also take all values in between. In between two different real numbers there is always some irrational number. Therefore, if the function were to take some other value it would have to take irrational values. Since it doesn't, then $5$ is the only value that it takes. – OscarRascal Jan 24 '20 at 17:01
  • Hint: $f(x)$ must be rational for all $x$ in the domain, including irrational $x$. – WaveX Jan 24 '20 at 17:02
  • @WaveX The rationality or irrationality of the inputs is irrelevant. – OscarRascal Jan 24 '20 at 17:04
  • @OscarRascal Got it Thank you – user744725 Jan 24 '20 at 17:06
  • @OscarRascal my mistake, I misinterpreted as it being a linear function rather than any continuous function, which makes my hint invalid indeed – WaveX Jan 24 '20 at 17:06

1 Answers1

5

The image of a continuous real function under a closed interval (like $1\leq x\leq3$) is also a closed interval. (This is a combination of the intermediate value theorem and the extreme value theorem.) The only closed intervals that don't contain any irrational numbers are singleton sets where the only value is rational. Therefore, since $5$ is in the range, the function must be $f(x)=5$ for all $1\le x \le 3$ and thus $f(1.5)=5$.