Discontinuity of second kind versus 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 with second kind discontinuity point not verifying the intermediate value property.