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# Math Help! Solution by tomorrow por favor! Thanks for the help!

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Hi, I have a new problem:

Let $$f:\mathbb R \to \mathbb R$$ be a function such that for any irrational number $$r$$ and any real number $$x$$, we have $$f(x)=f(x+r)$$. Show that $$f$$ is a constant function.

Again, please provide a full solution so i may understand how to do it.

Thanks for the help! Much appreciated.

Nov 15, 2018

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$$\text{The problem is to show that }f(x) = f(x+q),~q \in \mathbb{Q}\\ r \in \mathbb{R}-\mathbb{Q} \Rightarrow q-r \in \mathbb{R}-\mathbb{Q}\\ \text{i.e. }q-r \text{ is irrational}\\ f(x) = f(x+q-r)\\ x+q-r \in \mathbb{R} \\ f(x) = f(x+q-r) = f(x+q-r + r) = f(x+q)$$

$$\text{so }f(x) = f(x+q)=f(x+r),~q \in \mathbb{Q},~r \in \mathbb{R}-\mathbb{Q} \\ \mathbb{Q} \cup \left( \mathbb{R}-\mathbb{Q} \right) = \mathbb{R}, \text{ thus}\\ f(x) = f(x + y),~\forall y \in \mathbb{R},~\text{ i.e. }f \text{ is constant}$$

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Nov 15, 2018
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Thank you so much!

Nov 15, 2018