A sequence of continuous functions on $[0,1]$ which converge pointwise a.e. but does not converge uniformly on any interval 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?

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.
