I am finding lots on here about 'uniform convergence vs pointwise convergence' of a function but not the comparison for local uniform convergence.
It somehow intuitively seems to me that pointwise convergence of a function implies local uniform convergence given the correct interval (that you use to make it local).
Could someone please help explain this. Or give an appropriate example?
Thanks!
Pointwise convergence does not imply locally uniform convergence. If the domain of the sequence of functions is compact, locally uniform convergence is even equivalent to uniform convergence.
For example, consider the sequence of functions $f_n(x) = x^n$ on $[0, 1]$. This sequence converges pointwise to the function $f(x) = I\{x = 1\}$, but it does not converge uniformly in any neighborhood of $1$.
In the case of pointwise convergence at some point $x$, given $\epsilon>0$ and $f$ the limit function, you can find an $N(\epsilon,x)$ such that $|f_n(x)-f(x)|<\epsilon$ as long as $n>N(\epsilon,x)$.
And what about uniform convergence on a set $E$? First of all, pointwise convergence everywhere must be satisfied. So for each $x\in E$, we have an $N(\epsilon,x)$, and uniform convergence requires that whatever $x$ is, there will be a common $N(\epsilon)$ that suits all $x\in E$, namely, that $$\sup_{x\in E} N(\epsilon,x)<+\infty,$$ which is the essence of what uniform convergence says.
If $E$ is finite it is clear that pointwise convergence is just the same thing as uniform convergence. Otherwise, pointwise convergence tells nothing about uniform convergence. Locally uniform convergence isn't much different since it requires uniform convergence on a small neighbourhood ("how small" being subject to $x$, but still a finite-size thing) which may well contain infinitely many points and make the $\sup$ infinite.
