Complexity of approximating a real function using queries

Consider the following computational problem, where $I$ is the real interval $[-1,1]$:

There is a monotonically-increasing function $f: Ito I$. You are allowed to access it only through queries of the kind: "Given $xin I$, what is $f(x)$?". Let $x_0$ be an element of $I$ such that $f(x_0)=0$ (if it exists). Your goal is to find a value $x$ such that $|x-x_0|<epsilon$. How many queries do you need, as a function of $epsilon$?

All real numbers have infinite precision, as in the Real RAM model. It is allowed to do arbitrary computaitons on such real numbers – the only costly operations are the queries.

Here, the solution is simple: using binary search, the interval in which $x$ can lie shrinks by 2 after each query, so $log_2(1/epsilon)$ queries are sufficient. This is also an upper bound, since an adversary can always answer in such a way that the possible interval for $x$ shrinks by at most 2 after each query.

However, one can think of more complicated problems of this kind, with several different functions and possibly different kinds of queries.

What is a term, and some references, for this kind of computational problems?

_{Related posts in other sites:}

• _{First posted on cs.stackexchange.}
• _{This MathOverflow post is related, but there it is required to use a finite number of registers with a finite precision.}