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# Let I be an interval and let f : I → R be a continuous function such that f$I$ ⊂ Q. Show $in symbols$ that f is constant.

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## Answer to a math question Let I be an interval and let f : I → R be a continuous function such that f$I$ ⊂ Q. Show $in symbols$ that f is constant.

Murray
4.5
To show that the function f is constant, we can use a proof by contradiction.

Assume that f is not constant. This means that there exist two distinct points a, b ∈ I such that f$a$ ≠ f$b$.

Since f is continuous on the interval I, by the Intermediate Value Theorem, for any value y between f$a$ and f$b$, there exists a point c ∈ $a, b$ such that f$c$ = y.

Since f$I$ ⊂ Q, this means that for any y between f$a$ and f$b$, f$c$ = y must be a rational number.

Now, consider the real numbers between f$a$ and f$b$, which include both rational and irrational numbers. By the density property of the real numbers, we can always find an irrational number z between any two distinct rational numbers.

However, this means that there exists an irrational number z between f$a$ and f$b$ such that there is no point c ∈ $a, b$ such that f$c$ = z.

This contradicts the fact that f is continuous on the interval I.

Therefore, our assumption that f is not constant must be false, and thus f is constant.

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