Here's a classic result from any first course in mathematical analysis.

Intermediate Value Theorem. Let f:[a,b]-->R be a continuous function, for real numbers a,b, and suppose that f(a) < 0 < f(b). Then there exists c in (a,b) such that f(c)=0.

To understand what the theorem means it helps to sketch a graph of the function.

|     .-.
|    /   \         
|   /     
| /
What's happening is that the function starts off below the x-axis and then rises above it. Since the function is continuous (intuitively the graph is unbroken) we strongly expect that the function must hit the x-axis at some point in the interval.

Proving little theorems like this one is how we learn what mathematical rigour is. As we try to prove interesting results and make progress we are guided by our intuition about what should be true, but we must be able to produce detailed closely argued proofs that follow from the axioms. See the proof of the Intermediate Value Theorem for an object lesson.

A simple corollary of the theorem is that if we have a continuous function on a finite closed interval [a,b] then it must take every value between f(a) and f(b). To prove this, if v is such an intermediate value, consider the function g with g(x)=f(x)-v, and apply the IVT to g.