A function that can be "extended" to another function What do they mean when they say that a function can or cannot be "extended" to another function? I don't quite understand the terminology here. Whar does "extended" mean in this scenario? I want to show that a bounded continuous function cannot always be extended to a continuous function, but I don't know what this means. 
Can someone please clarify on this?
Thank you.
 A: Sometimes we say something like $\sum_{n=0}^{\infty}x^n$ can be extended to $\frac{1}{1-x}$. As opposed to $$\sum_{n=0}^{\infty}x^n= \frac{1}{1-x}$$ because the expressions on both sides of the equal sign have different domains. How can things be equal if they have different properties? 
So in this sense we can say $f$ defined on a domain $d$ can be extended to $g$ on domain $D$ a superset of $d$ when 
$g(x)=f(x)$ for all $x\in d$. I don't think (and could be wrong) that "extended" really has any other meaning unless stated. 
So for example let $f(x)=\sum_{n=0}^{\infty}x^n$ and then let $g(x)=\begin{cases} \frac{1}{1-x} \mbox{ when } |x|<1 \\ h(x) \mbox{ when } |x|\ge 1  \end{cases}$. 
So then $g(x)$ should be called an extension of $f$ regardless of how we pick $h$ but then people seem to use different adjectives to describe these extensions. Taking $h(x)=1/(1-x)$ might be called the natural extension. People use terms like "analytically" extended or "continuous" extended to describe more precisely the extension they are looking for. 
