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?
Sequence of Continuous Functions Converging Pointwise but Not Uniformly – Real Analysis
real-analysis
Best Answer
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.