What is the definition of a problem In computation theory, when talking about the computability and complexity of a
problem, what is the definition of a problem? 

How specific should a problem be? For example, can the followings
all be function evaluation problems?


evaluate $f$, where $f(x)=x^2, x \in \mathbb R$
evaluate any function in $\mathbb R^{\mathbb R}$.
evaluate any function in $Y^X$, where $X$ and $Y$ are any two sets.

Without restriction to the computability and complexity of a problem (or
even without restriction to computation theory), how is a problem defined? 

Can the above examples in 1 all  be  function evaluation problems?


Thanks.
A problem is anything you're trying to solve computationally. Problems typically specify an input and a desired output which, for most models of computation, will both be finite. Any statement of the form, "Given $X$, compute/evaluate/find/determine/decide whether/..." is a problem.

So, yes, all the things you describe in the question are problems. (Though, in the case where the input is the uncountable set&nbsp;$\mathbb{R}$, you need to be careful about how you specify what the input is.)