Intermediate Value Theorem

As long as we are dealing with a continuous function whose graph has no gap in it, the theorem works wonderfully in determining the existence of a zero. That is to say that, without the continuity requirement, the Intermediate Value Theorem breaks down. Hence, any solution claiming the existence of a zero must mention that the function is continuous before the theorem is invoked.
 

 
The example above attempts to show that the given polynomial has a zero between x = 0 and x = 1. What may appear obvious from the graph needs to be algebraically confirmed. This is accomplished by evaluating the polynomial at x = 0 and x = 1. Since every polynomial function is continuous throughout its domain, it follows from the theorem that the function must have at least one zero between x = 0 and x = 1.

To repeat, the continuity requirement is critical. Without it, the Intermediate Value Theorem breaks down.