The continuous functions $x^n$ on $[0,1]$ converge pointwise to a discontinuous function.Suggest a slogan
For \(n\geq1\) let \(f_n\colon[0,1]\to\RR\), \(f_n(x)\coloneqq x^n\). Each \(f_n\) is continuous, and \(f_n\) converges pointwise to the discontinuous function \(f\) with \(f(x)=0\) for \(0\leq x<1\) and \(f(1)=1\). For \(x_k\coloneqq1-1/k\), the iterated limits differ: \[ \lim_{k\to \infty} \lim_{n\to\infty} f_n(x_k) = 0 \neq 1 = \lim_{n\to \infty} \lim_{k\to\infty} f_n(x_k) . \]