Derivatives of pointwise converging sequence of functions may not converge anywhere? On page 137 of Analysis by Lieb & Loss it is stated that

The derivatives of a pointwise converging sequence of functions need
  not, in general, converge anywhere.

I would have thought that if a sequence of functions converges to some $f(x)$ they will all converge to the same derivative, that is, I would expect they would converge to $f'(x)$. So why is this not the case?

It's not the case because $f_n$ can be very small although $f_n'$ is very large.
If you allow complex-valued functions an example is very simple.
Define $f_n:[0,1]\to\Bbb C$ by $$f_n(t)=\frac{1}{n}e^{in^2t}.$$
Then $|f_n|\le1/n$, so $f_n\to0$ uniformly. But $|f_n'|=n$, so $f_n'$ does not converge at any point.
An example with real-valued functions is not quite so simple, because if $f$ is real-valued and $f'$ is continuous and also large at every point then $f$ must be large somewhere. But it's still not hard. For example, if $f_n$ is as above let $g_{2n}=\Re f_n$ and $g_{2n+1}=\Im f_n$. Then $g_n\to0$ uniformly but $g_n'$ converges nowhere.
