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.
| .-.
| / \
| /
a--/------b-------
| /
|/
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.