Can continuity of real functions be "globally" characterized? Most characterizations of pointwise continuous functions defined on an interval rely on "local" properties. That is, a function is continuous at $x_0 \in I$ if it satisfies some property (epsilon-delta, sequential, oscillation, etc); a function is continuous on an interval if it is continuous at all $x \in I$. 

Is there a characterization of pointwise continuous functions $f:(a,b) \to \mathbb{R}$ which relies solely on "global" properties of the function? 

An example (but, of course, incorrect, in this case) of a "global" property would be the intermediate value property. 
 A: Definition: $f$ is said to be continuous when pre-image of open sets are open sets, that is, if $A$ is open then $f^{-1}(A)$ is open.
A: This is just to mention that "open sets" may be replaced by "open intervals" in the definition I mentioned (and which Renan R. cites in his answer). 
Simply put an open set $O$ is one for which given any $x\in O$ there is an open interval $I$ for which $x \in I \subseteq O.$ So if inverse images $f^{-1}(I)$ are each open for any $I,$ then by taking unions of intervals we get to the more "abstract" version that inverse images of open sets are open. Conversely, since open intervals are a special type of open set, we can go the other way.
Note: It is not the case that the inverse image of an open interval has to be a single open interval. Consider $f(x)=x^3-100x$ and the inverse image of the interval $(-1,1)$ under $f.$ This will consist of three disjoint open intervals, one near each of $0,10,-10.$ And there are likely other examples where the inverse image of a single interval can be more complicated than that, perhaps have infinitely many components, or even an arbitrary open set (I don't have examples for either of these situations).
