When do we say that s function is defined Can we say that $\frac{\sin x}{x}$ is defined at $x=0$? When is a function defined within its domain?
 A: A function is defined once a unique output of the codomain is associated to each input from the domain. 
In the case of sin x/x, we implicitly use a technique whereby we extend this function, as defined in the nonzero numbers, to 0 as well by assigning the limit as x approaches 0 to the point 0.
So while the expression is not defined, there is a unique way we can extend it to a continuous entire function. 
A: We do not say f(x) = sin x /x is defined at x=0, although we certainly can-and should-say lim f(0) ---> 1 by the Squeeze Theorem. The limit of a function can certainly exist at a point where the function has no value. 
To really understand this, you have to understand the precise definition of a function on it's domain. Consider the precise definition of a function: Let A and B be nonempty sets.A function F from A into B is a nonempty subset of the Cartesian product A x B = { (a,b) ={a,{a,b}| a is in A and b is in B} where no 2 different ordered pairs have the same first member. The set of all first members of the function is called the domain of the function and the set of all second members of the function is called the range of f. Therefore, a function is not defined at a point a in A iff there is no such ordered pair in f where f(a) = z where z is in the range of f.
A limit of a real valued function,however, is defined by open interval (a,b) in the real line R of the domain and a corresponding open subset of the range (c,d)= (f(a).d) such that if |a-b| < r where r is some positive real number, then there exists some positive real number r' such that |c -d|< r'. This is certainly true even if d is not in the range of f.     
That answer your question?  
