TITLE: A sequence of continuous functions on $[0,1]$ which converge pointwise a.e. but does not converge uniformly on any interval
QUESTION [12 upvotes]: How to construct a sequence of functions that are defined and continuous on $[0,1]$ and it converges to zero a.e. but on any interval it does not converge uniformly?

REPLY [12 votes]: See example 7.4, page 77, in Counterexamples in Analysis, Gelbaum and Olmsted (the nice diagram on page 79 gives the essential idea of the construction). 
There, a sequence of continuous functions is constructed that converges to the function $f(x)=\cases{1/q,&$x=p/q$ in lowest terms, $p$ and $q$ integers with $q>0$\cr 0,&$x$ irrational}$.
That  the convergence is not uniform on any interval follows from the fact that $f$ is discontinuous at every rational $x$, and the fact that a uniform limit of continuous functions is continuous.