I am well acquainted with the concepts of pointwise and uniform convergence, namely on the former one fixes a point $x$ and then investigates if a given sequence of functions converges. In the latter, one sees if all $x$ can converge "at once". A more intuitive way to frame the latter is one can construct an $\epsilon$-tube around $f$ that contains all functions $f_n$ for all $n \geq N$ for some $N$ (depending on $\epsilon$ of course).
A friend of mine is taking analysis for the first time asked for an analogous image, but instead where pointwise convergence holds yet uniform convergence fails and I fell short. I am hoping someone on here can come up with a better mental picture than I can.
Best Answer
A canonical example, mentioned in the comments, is $f_n(x)=x^n$ on $[0,1]$. Converges pointwise to $0$ for $x<1$ and to $1$ for $x=1$. The convergence cannot possibly be uniform because a uniform limit of continuous functions is continuous.
A more illustrative example, in my opinion, is a "travelling bump". Let $f_n=1_{[n,n+1]}$. That is, $$ f_n(x)=\begin{cases}1,&\ n\leq x\leq n+1\\[0.3cm] 0,&\ \text{otherwise}\end{cases} $$ Then $f_n\to0$ pointwise, since for any $x$ we will eventually have $x<n$ for $n$ large enough. Meanwhile $\max\{|f_n(x)|: x \}=1$ for all $n$; that is $\operatorname{dist}(f_n,0)=1$ for all $n$, so the convergence is not uniform. Of course the example can be easily modified to have $f_n$ continuous or even smooth.