Real-valued function and extended-real-valued function, what's the difference? This might be a simple question, but  it is really confusing. I have not found any explicit definition or example of extended real-valued function. 

My guess is that f defined by f: x -> 1/x on (0,1) is REAl-valued, while g defined by g: x -> 1/x on [0,1) is EXTENDED REAL-valued. Is this the case?

Could someone please give some examples?
In general, an $E$-valued function is a function that takes values in $E$. 

The extended reals are $\mathbb{R}\cup\{\pm \infty\}$. So an extended-reals-valued function takes values that are either in $\mathbb{R}$, or possibly $\pm \infty$.
The only difference between Real valued function and Extended Real valued function is interval of domain . . .
In case of Real Valued Function Domain is (-∞,+∞) and in case of Extended Real Valued Function Domain is [-∞,+∞].
Now another Question arises that ∞ is a real number?
absolutely no . . .
But in mathematical analysis and calculus ∞ play an important role that's why it is named as extended real numbers , moreover it also satisfies some laws of binary operation (i.e addition and multiplication)
just as follows
5 + ∞ = ∞ + 5 = ∞
5 * ∞ = ∞ * ∞ = ∞