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My teacher showed us this function and told us it was continuous at all non-$\mathbb{Q}$ points:

$$ f(x) = \begin{cases} x & \text{ if } x\in\mathbb{Q} \\\\ 0 & \text{ if }x\notin\mathbb{Q} \end{cases} $$

However, Wolfram MathWorld says the Dirichlet function, which is very similar, is discontinuous at all points. Why is one continuous and the other not?

t.b.
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Anonymous
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  • On the other hand, the second function listed on that MathWorld page is a good example of a function continuous at each irrational and discontinuous at the rationals. It's harder to prove that the reverse situation cannot occur. – Dylan Moreland Sep 06 '11 at 01:03
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    Here's an article for those interested in (but not familiar with) @Dylan's statement: http://www.maa.org/mathdl/MM/0025570x.di021166.02p0055m.pdf – Tyler Sep 06 '11 at 01:11
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    Please. Read this: Wikipedia:Manual of Style/Mathematics/Why you should never use \mbox within Wikipedia This does not apply ONLY to Wikipedia. – Michael Hardy Sep 06 '11 at 01:27
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    The function your teacher showed you is not continuous at all irrationals (nor is it discontinuous at all rationals). It is, however, a nice example of a function which is continuous only at a single point (namely 0). – Ilmari Karonen Sep 06 '11 at 10:59

3 Answers3

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As other answers have Henning's answer has explained already, your teacher is wrong. However, my guess is that s/he was confusing your function with this related one: $$ f(x) = \begin{cases} 0, &\text{ if $x$ is irrational}, \\\\ 1/b, &\text{ if $x = a/b$ with $\gcd(a,b)=1$}. \end{cases} $$ This function does have the property that it is continuous at all irrational points, and discontinuous at the rationals.

Source: Look at the "modified Dirichlet function" $D_M(x)$ in the Mathworld article on the Dirichlet function.

Edit: It turns out that @lhf posted the same answer independently, but he links to the Wikipedia page of the function: click here.

Terminology: I just learned1 that this function is usually called Thomae's function, and not the modified Dirichlet function. I have known this example for some time, but not by any specific name. Wikipedia lists a number of other interesting names as well: the Riemann function, the popcorn function, the Stars over Babylon, the raindrop function, and the ruler function.

1In the post what functions or classes of functions are Riemann non-integrable but Lebesgue integrable, Hans Lundmark's comment (under Jonas Meyer's answer) gives the name of the function and the wikipedia link. Thanks to Theo Buehler for sharing the post.

Community
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Srivatsan
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  • That's a very nice answer now! However, I don't really deserve any credit, but rather Hans Lundmark, see the comment thread to Jonas's answer here. – t.b. Sep 06 '11 at 01:47
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Your teacher is wrong. Every neighborhood of any real number contains both rational and non-rational points, so the function is continuous at 0 only.

More specifically, let $\alpha$ be a positive irrational number (in particular $\alpha\ne0$, and the argument for negative irrationals is almost the same). Let's check whether $f$ is continuous at $\alpha$. For this to hold, then for every $\epsilon>0$ there must be a $\delta$ such that $$ |x-\alpha|<\delta \Rightarrow |f(x)-f(\alpha)|<\epsilon $$ Since $f(\alpha)=0$, the right-hand side of this is equivalent to $|f(x)|<\epsilon$.

As I'm going to prove that $f$ is not continuous at $\alpha$, I have the right to select $\epsilon$, and then I must prove that there's no $\delta$ that works for it. I choose $\epsilon=\alpha/2$. Now, for every possible positive $\delta$, the interval $(\alpha, \alpha+\delta)$ is open and therefore contains at least one rational number, which we can call $R$. Then, setting $x=R$ we get $|R-\alpha|<\delta$ (by construction), but $|f(R)|=R>\alpha$ is certainly larger than $\epsilon$, which was $\alpha/2$. Thus, $\delta$ fails to work, as promised.

hmakholm left over Monica
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  • Another way is to use the fact that a function $f$ is continuous at $x$ iff for all sequences $x_n \to x$, the sequence $f(x_n) \to f(x)$. Any $x$ can be approached by both rational and irrational sequences $x_n$; the corresponding limits of $f(x_n)$ then are equal to $x$ and $0$ respectively. For continuity, we require $x = 0$. – Srivatsan Sep 06 '11 at 01:08
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    That works, too -- but for the purpose of convincing an errant teacher to mend his ways, I think it is useful to go as close to the definitions as practically possible. (I think epsilon-delta definitions are more common at the elementary calculus level than sequential continuity, but I may be wrong in the instant case, of course). – hmakholm left over Monica Sep 06 '11 at 01:31
  • You are right, I do not think they do sequential continuity in the elementary calculus classes. I just pointed it out for completeness sake; I am not claiming any one way is easier than the other. – Srivatsan Sep 06 '11 at 01:39
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Perhaps your teacher meant Thomae's function, which is continuous at all irrational numbers and discontinuous at all rational numbers.

lhf
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