Intermediate Value Theorem
Key Questions

To answer this question, we need to know what the intermediate value theorem says.
The theorem basically sates that:
For a given continuous function#f(x)# in a given interval#[a,b]# , for some#y# between#f(a)# and#f(b)# , there is a value#c# in the interval to which#f(c) = y# .It's application to determining whether there is a solution in an interval is to test it's upper and lower bound.
Let's say that our
#f(x)# is such that#f(x) = x^2  6*x + 8# and we want to know if there is a solution between#1# and#3# (in the#[1,3]# interval).
#f(1) = 3#
#f(3) = 1#
From the theorem (since all polynomials are continuous), we know that there is a#c# in#[1,3]# such that#f(c) = 0# (#1 <= 0 <= 3# )//Hope it helps.

Answer:
There are several definitions of continuous function, so I give you several...
Explanation:
Very roughly speaking, a continuous function is one whose graph can be drawn without lifting your pen from the paper. It has no discontinuities (jumps).
Much more formally:
If
#A sube RR# then#f(x):A>RR# is continuous iff#AA x in A, delta in RR, delta > 0, EE epsilon in RR, epsilon > 0 :# #AA x_1 in (x  epsilon, x + epsilon) nn A, f(x_1) in (f(x)  delta, f(x) + delta)# That's rather a mouthful, but basically means that
#f(x)# does not suddenly jump in value.Here's another definition:
If
#A# and#B# are any sets with a definition of open subsets, then#f:A>B# is continuous iff the preimage of any open subset of#B# is an open subset of#A# .That is if
#B_1 sube B# is an open subset of#B# and#A_1 = { a in A : f(a) in B_1 }# , then#A_1# is an open subset of#A# . 
Answer:
It means that a if a continuous function (on an interval
#A# ) takes 2 distincts values#f(a)# and#f(b)# (#a,b in A# of course), then it will take all the values between#f(a)# and#f(b)# .Explanation:
In order to remember or understand it better, please know that the math vocabulary uses a lot of images. For instance, you can perfectly imagine an increasing function! It's the same here, with intermediate you can imagine something between 2 other things if you know what I mean. Don't hesitate to ask any questions if it's not clear!