Pointwise vs. Uniform Convergence

Solution 1:

Comparison

Pointwise convergence means at every point the sequence of functions has its own speed of convergence (that can be very fast at some points and very very very very slow at others).

Imagine how slow that sequence tends to zero at more and more outer points: $$\frac{1}{n}x^2\to 0$$

Uniform convergence means there is an overall speed of convergence.

In the above example no matter which speed you consider there will be always a point far outside at which your sequence has slower speed of convergence, that is it doesn't converge uniformly.

Another Approach

One can check uniform convergence by considering the "infimum of speeds over all points". If it doesn't vanish then it is uniformly convergent. And that gives another characterization as the ones with nonvanishing overall speed of convergence.

Solution 2:

It may help if you unfold the definition of limit in pointwise convergence.

Then pointwise convergence means that for each $x$ and $\epsilon$ you can find an $N$ such that (bla bla bla). Here the $N$ is allowed to depend both on $x$ and $\epsilon$.

In uniform convergence the requirement is strengthened. Here for each $\epsilon$ you need to be able to find an $N$ such that (bla bla bla) for all $x$ in the domain of the function. In other words $N$ can depend on $\epsilon$ but not on $x$.

The latter is a stronger condition, because if you have only pointwise convergence, it may be that some $\epsilon$ will require arbitrarily large $N$ for some $x$s.

For example, the functions $f_n(x)=\frac{x}{n}$ converge pointwise to the zero function on $\mathbb R$, but do not converge uniformly. For example, if we choose $\epsilon=1$, then the convergence condition boils down to $N>|x|$. For each $x\in\mathbb R$ we can find such an $N$ easily, but there's no $N$ that works simultaneously for every $x$.

Solution 3:

$f_n\to f$ pointwise on $(a,b)$ if for each fixed $x\in(a,b)$, $|f_n(x)-f(x)|\to 0$ as $n\to\infty$. Notice this is a pointwise (local) criterion.

On the other hand, $f_n\to f$ uniformly on $(a,b)$ if $\sup_{a< x<b}|f_n(x)-f(x)|\to 0$ as $n\to\infty$. This is a "global" criterion in that is requires the maximum of all the pointwise errors on $[a,b]$ to tend to zero.

As an example, $f_n(x)=x^n$, $0\le x\le 1$ converges pointwise to $f(x)=\begin{cases} 0, &0\le x<1,\\ 1, &x=1.\end{cases}$, because the first condition above holds, but the convergence is not uniform since the second condition does not hold.

Solution 4:

The pointwise convergence depends on each $x$, that is you need to count and fix every $x$ before passing to the limit. On the other hand, uniform convergence is independent of $x$. Hence, the pointwise convergence is enough when your space $X$ is a countable set and not enough when you are working with an uncoutable set like $[0,1]$, in such case you need a uniform convergence to generalise your convergence on $[0,1]$.