The Intermediate Value Theorem states that if a function f(x) is continuous on [l, r]; then for any number L within f(l) and f(r), there exists a number a in [l, r] such that f(a)=L

This statement might seem complicated when you first read it, but the basis of it is simple to explain.

Imagine you want to travel from Vancouver BC, to Seattle, USA by car. The only way you can do it is by crossing the border. So, even if you travel all the way to Toronto or Montreal, if you want to go to Seattle you must cross the border. So, at some point during your trip you have to be at the border. This is the basis of the intermediate value theorem (IVT).

The function in our example is the trajectory of the car, which is continuous from Vancouver (l) to Seattle (r). Between your position in Vancouver ( f(l) ) and Seattle ( f(r) ), you must cross the border (L).

This argument relies explicity on the concept of continuity because the first assumption of the  IVT is that the  the function has to be continuous between [l, r].