Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

While learning a little Fourier analysis, I ran into this interesting phenomenon:

Consider a series of sawtooth waves such that the height and width of the sawteeth shrinks to zero, but the slope of the sawteeth remains the same. To be specific, let

$$f_n(x) = \frac{nx - \lfloor nx\rfloor}{n}$$

Then define

$$F(x) = \lim_{n\to\infty}f_n(x)$$

It seems intuitively clear that $F(x) = 0$ for all $x$ because the global maximum of $f_n$ is $\frac{1}{n}$.

If $F(x) = 0$, then we should have $F'(x) = 0$ as well. However, if we choose an irrational value of $x$, then $f'_n(x) = 1$ for all $n$, so if $F'(x)$ is found instead by taking

$$F'(x) = \lim_{n\to\infty}f'_n(x)$$

we do not get $F'(x) = 0$.

It seems like the derivative of a limit is not the same as the limit of a derivative, which is pretty counterintuitive to me.

What's going on?

share|cite|improve this question
This example illustrates the principle well: Functions can be close while their derivatives are not. The standard image to invoke is similar to what you have given here: one function can be constant, while a "nearby" function can stay close to that constant function while oscillating maniacally. Another example to consider would be $\frac{\sin(nx)}{n}\to 0$. – Jonas Meyer Sep 15 '11 at 22:44
That's correct, they are not the same. (In particular, $\lim f_n'(x)$ exists only when $x$ is irrational, and is then equal to $1$ - can you see why?) Since the derivative is itself a limit, this is also a special case of the fact you cannot (in general) interchange two limits. – anon Sep 15 '11 at 22:46
@Mark Eichenlaub : As it has been stated, I'd like to mention that there is a possibility that a lot of interesting things missing when we use convergence under a norm like the $\mathcal{L}^p$. – Rajesh Dachiraju Sep 16 '11 at 8:34
You may want to have a look at this as well, although a different, you might find it interseting. under the section 'Applications'. It talks about the differentiability and the derivative of the limit function under pointwise convergence. – Rajesh Dachiraju Sep 16 '11 at 8:38
up vote 2 down vote accepted

You say that you expect to be able to interchange limit and derivative operations. Now, derivative itself includes a limit operation, so I wonder whether you expect to be able to interchange any two different limit operations. If so, this discussion by Tim Gowers might be helpful.

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.