Does Thomae's Function has functional limit for all points in R?

Question

I know that Thomae's function is continuous at all irrational numbers, so there are functional limits for those point. However, I don't know about rational numbers...

Thank you!

Here read this. http://en.wikipedia.org/wiki/Thomae's_function It talks all about it. — user60887, Mar 04 '14 at 03:26
Thanks! However, I only see that talks about continuity but not functional limit? — Stephanie, Mar 04 '14 at 03:43

score 2 · Answer 1 · edited Apr 13 '17 at 12:21

2

If $a$ is a rational number, and $f$ is Thomae's function, then $\lim\limits_{x \to a} f(x) = 0$, but $f(a) \ne 0$. Thus Thomae's function has a limit at every real number, but is continuous exactly at the irrational numbers.

The fact that $\lim\limits_{x \to a} f(x) = 0$ for all $a$ is mentioned here; the proof is essentially identical to the irrational case, which is proven here.

edited Apr 13 '17 at 12:21

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answered Mar 31 '14 at 17:26

Caleb Stanford

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Does Thomae's Function has functional limit for all points in R?

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