What is the definition of limit superior/inferior of real function? http://en.wikipedia.org/wiki/Upper_limit

Wikipedia has the definition for metric space, but it doesn't have a definition for extended real-field.

It just says, 'Limit inferior and limit superior of a real function are always well-defined in $\overline{\mathbb{R}}$'.

In that sense, I don't think the definition for metric space covers real-valued functions. (Isolated points)

What's the definition it would be?
Say one considers a function $f:(0,+\infty)\to\mathbb R$ and one wants to determine the limsup/liminf of $f$ at $0$. Then one would define $g(t)=\sup\{f(x)\mid0\lt x\lt t\}$, $h(t)=\inf\{f(x)\mid0\lt x \lt t\}$, and
$$
\limsup\limits_{x\to0}f(x)=\inf\{g(t)\mid t\gt0\}=\lim\limits_{t\to0}g(t),
$$
and
$$
\liminf\limits_{x\to0}f(x)=\sup\{h(t)\mid t\gt0\}=\lim\limits_{t\to0}h(t).
$$
Example: $f(x)=\sin(1/x)$ for every $x\gt0$, then $g(t)=+1$ and $h(x)=-1$ for every $t\gt0$ hence $\limsup\limits_{x\to0}f(x)=+1$ and $\liminf\limits_{x\to0}f(x)=-1$. 

Similar but more involved: if $f(x)=(2+x^2\cos(1/x))\sin(1/x)$ for every $x\gt0$, then $\limsup\limits_{x\to0}f(x)=+2$ and $\liminf\limits_{x\to0}f(x)=-2$.
Just consider the function 

$$ f(x) = \frac{|x|}{x}.$$

Can you determine the limsup and the liminf as $x \to 0$ of this function? See here for the definition.