 A function f: R -> R is said to have period p if f(x) = f(x+p) for all real x. Show that a continuous function R -> R with period 1 and pi is constant. 

To show that a continuous function \( f: \mathbb{R} \to \mathbb{R} \) with periods 1 and \(\pi\) is constant, we need to use the fact that if \( f \) has two incommensurate periods (i.e., periods whose ratio is irrational), then the function must be constant. 

Let's proceed with the proof step by step:

1. **Definition of Periodicity**:
   Given \( f(x+1) = f(x) \) for all \( x \in \mathbb{R} \), \( f \) has period 1.
   Similarly, given \( f(x+\pi) = f(x) \) for all \( x \in \mathbb{R} \), \( f \) has period \(\pi\).

2. **Combining Periods**:
   We need to show that these periods imply \( f \) is constant. Let's examine the implications of having both periods 1 and \(\pi\).

3. **Rational and Irrational Periods**:
   Assume \( f \) is not constant. Then, there exist \( x_1, x_2 \in \mathbb{R} \) such that \( f(x_1) \neq f(x_2) \).

   Consider the set \( S = \{ n + m\pi \mid n, m \in \mathbb{Z} \} \). This set \( S \) is dense in \(\mathbb{R}\). This means for any real number \( x \) and any \(\epsilon > 0\), there exist integers \( n \) and \( m \) such that \( |x - (n + m\pi)| < \epsilon \).

4. **Continuity and Density**:
   Given that \( f \) is continuous, the value of \( f \) at any point is determined by its values arbitrarily close to that point. Since \( f(x) = f(x + n + m\pi) \) for all integers \( n \) and \( m \), the function takes the same value at any point that is in the dense set \( S \).

   Thus, by continuity, \( f \) must take the same value at all points in \(\mathbb{R}\).

5. **Conclusion**:
   Since \( f(x) = f(y) \) for any \( x, y \in \mathbb{R} \), we conclude that \( f \) is constant.

Therefore, we have shown that a continuous function \( f: \mathbb{R} \to \mathbb{R} \) with periods 1 and \(\pi\) must be constant.

Do you want details or have any questions?

Here are 5 more questions you might consider:

1. What is the significance of irrational periods in determining the constancy of a function?
2. How does the density of the set \( \{ n + m\pi \mid n, m \in \mathbb{Z} \} \) in \(\mathbb{R}\) play a role in the proof?
3. Can you provide an example of a function with a rational period that is not constant?
4. What happens if a function has two rational periods? Does the function need to be constant?
5. How would the proof change if we replaced \(\pi\) with another irrational number?

**Tip:** When dealing with periodic functions, examining their behavior with respect to rational and irrational periods can provide deep insights into their properties.

Explore how to prove that a continuous function with periods 1 and pi is constant. This problem delves into the properties of periodic functions and their implications in real analysis.

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