For given $f$ from reals to reals, the associated oscillation function is defined as follows:
$$\textstyle osc_f(x):= \lim_{n\rightarrow \infty} [\sup_{y \in B(x, \frac{1}{2^n}) } f(y)-\inf_{z \in B(x, \frac{1}{2^n}) } f(z) ] $$
Intuitively speaking, if $f$ is discontinuous at $x$, $osc_f(x)$ measures 'how big' the gap in the graph is.
The function $osc_f$ harks back to the days of Riemann, Hankel, and Ascoli. One great advantage is that, given $osc_f$, the formula
$f$ is continuous at $x$
is equivalent to $osc_f(x)=0$, where the latter is much simpler (from a logical pov).
My question is whether similar constructs are found in the literature (I could not find any, but they may be known by another name). For instance, let $B_n(f, x)$ be the $n$-th Bernstein polynomial for $f$ at $x$. Define the deviation of $f:[0,1]\rightarrow [0,1]$ at $x$ as follows:
$$\textstyle dev_f(x):= \lim_{n\rightarrow \infty} [\sup_{y \in B(x, \frac{1}{2^n}) } |f(y)-\lim_{m\rightarrow \infty} B_m(f, y) | ] $$ Intuitively speaking, if $f(x)$ does not equal its Bernstein approximation $\lim_{m\rightarrow \infty} B_m(f, x)$, then $dev_f(x)$ measures 'how much' the deviation between the two is. Perhaps one defines error terms in a similar way?
