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## Homework Statement

f:ℝ→ℝ is defined as f(x)= 2x if x is rational and f(x)=4-2x if x is irrational.

Is it true that lim x→1/2=1?

**2. The attempt at a solution**

Intuitively it seems that as x gets ever closer to 1/2 from either side that the function will oscillate between numbers very close to 1 and 3 and therefore there the limit doesn't exist.

Firstly is my intuition right?

Secondly how does one show this using the definition of a limit or some other theorem?

I thought about using the fact that all (x

_{n})[itex]\subseteq[/itex]ℝ such that (x

_{n})→1/2 would have to imply that (f(x

_{n}))→1. Again i can't formally show that this can't be.

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