Some examples of local and nonlocal properties Today I learned that continuity at a point is a local property. Concretely, if  $f:  \mathbb R \to \mathbb R$ is continuous on $[-K,K]$ for all $K \in \mathbb R$ then $f$ is continuous on $ \mathbb R$.
Uniform convergence on the other hand is not a local property: if $g_n \to g$ uniformly on $[-K,K]$ for all $K \in \mathbb R$ then it does not follow that $g_n \to g$ on $\mathbb R$. (it is not clear to me though if this is only  because $ \mathbb R$ is not compact and it would hold if $\mathbb R$ was compact)
Since I still don't fully grasp what a local property is and what a non-local property is I would like to kindly request you to post some examples of both to help me get a feel for it. 
Added For example, is differentiability a local property like contiuity? Does it hold that if  $f:  \mathbb R \to \mathbb R$ is differentiable on $[-K,K]$ for all $K \in \mathbb R$ then $f$ is  differentiable on $ \mathbb R$?
 A: Uniform continuity is also not local. A continuous function is uniformly continuous on compact sets. For example, the function $f(x)=x^2$ is uniformly continuous on any finite interval $[-K,K]$ but not on the whole real line. 
We want to theck the definition of uniform continuity, i.e. for each $\varepsilon>0$ there exists $\delta>0$ such that for all $|x-y|<\delta$, you have $|f(x)-f(y)|<\varepsilon$.  Equivalently, if $x_n,y_n$ are two sequences with $|x_n-y_n|\to 0$, then $|f(x_n)-f(y_n)|\to 0$. If you are inside a compact set the sequences $x_n,y_n$ can't do much, but if you are on the whole real line, you can take sequences $x_n,y_n\to \infty$ such that $|x_n-y_n|\to 0$. For example take $x_n= n+1/n$, $y_n=n-1/n$. Then we have
$$|f(x_n)-f(y_n)|= n^2+2+1/n^2-n^2+2-1/n^2=4 \nrightarrow 0$$ 
The local properties depend only on local data, but here if you look at the definition of uniform continuity you must able to check it for all $x,y\in \mathbb R$ with $|x-y|<\delta$, and the example shows how this can fail.  
