n-approximable functions I came across the following definition in a paper:


We can extend the notion of an $n$-c.e. [n-computably enumerable] set to a notion that measures the number of fluctuations of a function as folows: For every $n \geq 1$, call $f : N \rightarrow R$ n-approximable if there is a rational-valued computable approximation $\varphi$ such that $\lim_{k\rightarrow \infty} \varphi(x, k) = f(x)$ and such that for every $x$, the number of $k$’s such that $\varphi(x,k + 1) − \varphi(x,k) &lt; 0$ is bounded by $n − 1$. That is, $n − 1$ is a bound on the number of fluctuations of the approximation. 

Note that the $1$-approximable functions are precisely the lower semicomputable ($\Sigma_1^0$) ones (zero fluctuations). Also note that a set $A \subseteq \mathbb N$ is $n$-c.e. if and only if the characteristic function of A is n-approximable.


I have two questions. 


Is this a commonly used principle? Google didn't give me any relevant results.
If I have an $n$-c.e. function $\varphi(x)$, with $\varphi(x,k)$ approximating it. Couldn't I just build a Turing machine $\varphi'(x, k')$ that dovetails the computation of $\varphi(x, k)$ until it sees two consecutive halting machines with $\varphi(x, k+1) &lt; \varphi(x, k)$, stops the dovetailing, and outputs $\varphi(x, k+1+k')$ (ie. it computes $\varphi(\cdot, \cdot)$ but starting from k+1). Then, if $\varphi(x)$ is $n-c.e.$ (and not $n+1$-c.e.), it is also $n-1$-c.e., and by induction it is just approximable.


So the definition doesn't seem to make sense. Am I missing something?
This is a special case of what is called an $f$-c.e. set. A set $A$ is $f$-c.e. iff there is computable function $g$ such that 


$\chi_A(x) = \lim_i g(x,i)$, and 
$\forall x \ |\{s \mid g(x,i) \neq g(x,i+1) \}| \leq f(x)$. 


If we consider $f(x)=n$ then we get the definition of $n$-c.e sets.

They don't need to be c.e. because c.e. is not closed under complement. Take two c.e. sets and take their difference. It is a $2$-c.e. set but it does not need to be a c.e. set.

Barry Cooper and colleges have been interested in these sets and the notion of computability in limit. You may want to check his "Definability in the Real Universe" article from "Computability In Context: Computation and Logic in the Real World".