Suppose \((f_n)\) is a sequence of continuous functions on \([0,1]\) converging pointwise to a continuous function \(f\). Must \(f_n\to f\) uniformly? Prove your answer.
Solution
No. Define \[ f_n(x)=\max\{1-n|x-1/n|,0\}. \] Each \(f_n\) is continuous on \([0,1]\), and \[ \|f_n\|_\infty=f_n(1/n)=1. \] Hence \((f_n)\) cannot converge uniformly to \(0\).
On the other hand, \(f_n\to0\) pointwise. Indeed, \(f_n(0)=0\) for every \(n\). If \(x>0\) is fixed, then for all sufficiently large \(n\) one has \(2/n<x\), while \(f_n\) is supported in \([0,2/n]\). Thus \(f_n(x)=0\) eventually. Therefore \[ f_n(x)\longrightarrow0 \qquad(x\in[0,1]), \] and the pointwise limit \(f\equiv0\) is continuous.
So pointwise convergence of continuous functions to a continuous limit does not imply uniform convergence.