To show that if $g : \mathbb{R} \to \mathbb{R}$ is a continuous function that only takes values in $\mathbb{Q}$ (the set of rational numbers), then $g$ must be a constant function, we can use the following reasoning:

Proof by Contradiction

1. Assume $g$ is not constant:
   Suppose, for contradiction, that $g$ is not a constant function. Then there exist two points $x_1$ and $x_2$ in $\mathbb{R}$ such that: $g(x_1) \neq g(x_2)$ Since $g$ takes values only in $\mathbb{Q}$, this implies that $g(x_1), g(x_2) \in \mathbb{Q}$, and these values are distinct rational numbers.

2. Continuity of $g$:
   The function $g$ is given as continuous. By the intermediate value theorem (a fundamental result of real analysis), if a function is continuous on an interval and takes two different values at two points, it must take every value between those two points. This means that for any real numbers $a$ and $b$ with $g(a) \neq g(b)$, the function must take every value in the interval $(g(a), g(b))$.

3. Contradiction with the rationals:
   The rational numbers $\mathbb{Q}$ are not dense in the real numbers in the sense of continuity: there are irrational numbers between any two distinct rational numbers. Thus, if $g$ were to take every value between $g(x_1)$ and $g(x_2)$, it would need to take irrational values as well, contradicting the assumption that $g$ only takes values in $\mathbb{Q}$.

4. Conclusion:
   The only way for a continuous function to take values only in $\mathbb{Q}$, a set that is "sparse" in $\mathbb{R}$ (in the sense that it does not contain any interval), is for the function to be constant. Hence, $g$ must be a constant function.

Final Answer:

The function $g : \mathbb{R} \to \mathbb{R}$ that takes values only in $\mathbb{Q}$ and is continuous must be a constant function.

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Do you have any questions or want further details?

Here are 5 related questions:

1. What is the intermediate value theorem?
2. Can a function take irrational values and still be continuous?
3. What are some other examples of continuous but non-constant functions?
4. How do we define continuity for a function from $\mathbb{R} \to \mathbb{Q}$?
5. Can a function that only takes rational values ever be discontinuous?

Tip: The intermediate value theorem is a powerful tool in analysis, especially when working with continuous functions. Familiarizing yourself with its applications can help in solving a wide range of problems.