Discontiuity of second kind and Darboux property
A real function defined on a interval I is said to have the Darboux (or intermediate value) property if, whenever a and b are from I and c is any number between f(a) and f(b) there exists a number x (depending of c) such that f(x)=c.
Introducing the Darboux rectangle this reformulate as follows: whenever a and b are from I, each parallel segment to x'x axis in the Darboux rectangle intersect the graphic of f.
There exists discontinuous functions verifying the intermediate value property.