Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A function $t: [a, b] \rightarrow \mathbb{R}$ is called a step function when a $k \in \mathbb{N}$ and numbers $z_0,...,z_k$ with $a = z_0 \leq z_1 \leq ... \leq z_k = b$ exist, such that for all $i \in \{1,2,...k\}$ the restriction $t |_{(z_{i-1},z_{i})}$ is constant. Let $f: [0,1] \rightarrow \mathbb{R}$ be defined by $$f(x) = \left\{ \begin{array}{rcl} 1, & \mbox{if} & x \in \mathbb{Q} \\ 0, & \mbox{if} & x \notin \mathbb{Q} \end{array} \right.$$


(i) The function $f$ is a point-wise limit of step functions.

(ii) There is no uniform convergent series of step functions, whose limit-function is $f$.

So the book I'm reading mentions this Dirichlet function all the time. Still I'm having trouble finding a solution to this exercise. All help is very much appreciated!

share|cite|improve this question
up vote 2 down vote accepted

For part ii).

If a sequence of Riemann integrable functions $f_n$ converges uniformly to $f$, then what can you say about the Riemann integrability of $f$?

It seems like this question is trying to point out a drawback of Riemann integration which, if I remember correctly, motivated the discovery of Lebesgue integration.

share|cite|improve this answer

A hint for the first part would be to recall that (a) the rationals are countable, so it would take a countable number of "steps" to "cover" them (since the irrationals are dense, you'll need to drop to 0 "in between" each rational number), and that (b) one could increase the number of steps (but decrease the size of each step) for each function in your sequence.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.