Picture a curve, maybe f(x)=x^2.  Regardless of how much you zoom in
on any part of the curve, the values form a continous curve.  Pick any
point on the curve, and the infinite number of values between whatever
you arbitrarily choose as a minimum and maximum value for x will form
a coninous curve.  In the given example, where f(x)=1 when x>= 0 and
f(x)=0 when x<0, if you get close enough, there is a disjunction rught
when you move left of 0.  The line has to break.  To be a contiguous
line, there would have to be intermediate values of x which would
result in all of the values of y between 0 and 1.  There are no such
values in the sample function.  You have to have a 1:1 mapping for any
y=f(x) between yMAX and yMIN - referring to a continuous function as a
continuous mapping, as done on
http://mathworld.wolfram.com/ContinuousFunction.html, might help some.

--Danny

Bill wrote regarding '[OT] When is a function not continuous?' on Mon, Dec 19 at 13:31:
> In order to study up on my electronics, I've had to take a detour and 
> brush up on calculus (there seems to be an infinite number of rabbit 
> trails to have to chase down on the way to "success.")   But I'm 
> determined, so I bought a used book called QUICK CALCULUS, Second 
> Edition, by Daniel Kleppner and Norman Ramsey.  Found it for a couple of 
> bucks in the local used book shop.  It has a lot of notes in the 
> margins, but that's OK with me.
> 
> I've gotten the preliminary stuff out of the way (functions, graphs, 
> trig, exponentials and logs) and started on the differential calculus 
> part.  The first section there is "limits," and I've hit a snag in my 
> thinking.  Given a function f(x), such that f(x) = 1 for x .GE. 0, and 
> f{x} = 0 for x .LT. 0, is that a continuous function or a discontinuous 
> function?
> 
> Since for every possible real number x there is a number f(x) which is 
> either zero or one, it seems to be that it would be continuous.  The 
> book says it is discontinuous, though.  Since they also say "A more 
> picturesque description of a continuous function is  that it is a 
> function you can graph without lifting your pencil from the paper in the 
> region of interest," though, I can see where it might possibly be 
> discontinuous, but it is against intuition.  I think that possibly they 
> meant to write  "f(x) = 1 for x .GT  0" rather than using the .GE.
> 
> Any thoughts would be appreciated.  (I'm going to read over  this email 
> carefully before sending it, to make sure I don't sound like English is 
> my 27th language.  :-)
> 
> Thanks,
> Bill
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