Function

1. Yes, it is a discontinued function.
One distinct point at (0,0) and a continuous curve at [1,inf)

2. No. y = ln x valid only for x belongs to (0,inf).
There is a lack of functioned values for x belonging to R\(0,inf) = (-inf,0]

3. No.
(i) Lack of functioned values for R\[-1,1]
(ii) More than one* functioned values for each x within [-1,1]

Remarks:
* This depends on the definition of f(x).

What we have to bear in mind is that, a function must give one and only one functioned value within the range for every x which belongs to the domain.

2011-09-07 14:14:28 補充：
For each function, domain must be "filled up" (Each x has a y assigned)
But the range does not needed to be "filled up".
The set of functioned value is called "co-domain", which must be a subset of range.

2011-09-07 14:16:12 補充：
For Question 1, the function is defined explicitly and the domain is nothing but {0}U[1,inf) the range is R.

2011-09-07 14:18:03 補充：
For Question 2 and 3, it involves the concept of Inverse Function.
For a function to have inverse, it must be bijective, which means
(i) one-one (injective); and
(ii) onto (surjective).
exp(x) is bijective and sin(x) is not.