learning about the oscillation of a function. Let $a \in Q \subseteq \mathbb{R}^n$.  for given $\delta > 0$, let
$$ A_\delta = \{ f(x) : x \in Q \; \; \text{and} \; \; |x-a|<\delta \} $$
and $M = \sup A_\delta $ and $m = \inf A_\delta $. My book defines oscillation of $f$ at $a$ as
$$ \nu(f,a) = \inf_{\delta > 0 } ( M - m) $$
But, my teacher wrote in class as
$$ \nu(f,a) = \lim_{\delta \to 0^+} (M-m) $$
I am confused trying to see if they are equal, or maybe it is a typo. I need some help trying to understand this concept. thanks
 A: The two definitions are indeed equivalent for bounded functions on $Q$. Each definition has its own advantage.
The problem with defining $\nu$ in terms of a limit is that in order to use the quantity, you need to know that the limit exists.  The advantage of your book's definition, then, is that it doesn't depend on a limit.  If we say that
$$
\nu(f,a) = \inf_{\delta > 0 } ( M - m)
$$
Then we're specifying that we're looking for the greatest lower bound of all $M-m$ over all possible $\delta$.  We're not saying which $\delta$ will give us that lower bound, nor are we implying the existence of any sort of limit as $\delta \to 0^+$.
Now, it ends up that for a bounded function $f$, the limit $\lim_{\delta \to 0^+}(M-m)$ exists.  Moreover, $\lim_{\delta \to 0^+}(M-m)$ is necessarily smaller than any $M-m$ for any choice of $\delta$.  So, after stating that this limit exists, it makes sense to say that
$$
\lim_{\delta \to 0^+} M-m = \nu(f,a) = \inf_{\delta > 0 } ( M - m)
$$
The advantage of using the limit definition is that this definition explicitly tells you that you can find this quantity by making $\delta$ arbitrarily small. 
Since we only ever talk about taking the (proper) Riemann integral of bounded functions, these two definitions are equivalent.  Another definition that may be used is
$$
\nu(f,a) = \liminf_{\delta \to 0^+} M-m
$$
I hope that clears things up.
