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The meaning of Continuity?

  1. May 30, 2003 #1
    Is it just me or is it the text didn't explain well enough about continuity? I totally don't understand it. Not only that, when it covers about derivatives it first introduced another defination of slope. Then it says that the derivative of x^2 is 2x. How does slop relate to derivative?

    Can someone explain what continuity, and discontinuity is?
  2. jcsd
  3. May 30, 2003 #2
    What text are you using?

    Read several definitions in different texts, they'll be pretty similar, but the wording may be better in some. The slope is generally defined as the tangent line at a point, and is equal to the derivative at that point. And don't be afraid to play around with it. It won't bite you.
  4. May 30, 2003 #3
    I'll explain to you how slope relates to the derivative, and hopefully in the process give you an idea about continuity:

    The derivative of a function at a point represents the slope of the function at that point.

    For a linear equation of the form:

    y = mx + b

    we know from high school that the slope of the graph is given by m. The greater the magnitude of m, the steeper the slope and vice versa. This slope is valid for any point on the graph, as it is constant. Now the next question that we want to ask is how do you calculate the slope of a curve? Here the slope is not constant, it is dependant on where you measure it.

    Lets look at the function y = x2. Suppose I pick a point x and want to measure the slope of the curve at that point. Well lets look at the tangent line to the curve at that x. That is the line that touches the curve at only the point x. Why the tangent line? Well the tangent line is a straight line, and we know how to measure the slope of a striaght line, (we did that above). And if we make the straight line "small" enough then we can shrink it so that it becomes a point, the point x! So we can talk about the slope of the curve at the point x. This of course assumes that we can shrink the line to a point and still measure the slope of the line, (which we is talked about later).

    We can find the slope of the tangent line if we know two points on the line. Well we know one point, (x, y(x)). Now lets pick a point a small distance away from x, say (x+h), where h is a small arbitrary constant. Well now we know another point ((x+h), y(x+h)). We can now calculate the slope of the tangent line:

    m = (y(x+h) - y(x))/((x+h - x) = ((x+h)2 - x2)/h

    m = (x2 + 2xh + h2 - x2)/h

    m = 2x + h

    We now know the slope of the tangent line to y at the point x. We didn't actually specify h, other than to say it was small and arbitrary. So lets let h = 0, then:

    m = 2x

    Voila! We have the derivative of y at the point x, which is equal to 2x. The derivative representing the slope of y at the point x. So now you can see how the slope of y changes with respect to x.


    The above is not a rigorous proof for finding the derivative. The actual proof relies on limits, which is why I made the point earlier that we had to assume it made sense to talk about calculating the slope of a line as it shrank to zero length, (ie. became a point). I'm not sure if you've talked about limits yet, so I won't go into them here.

    Continuity is an important property. Basically it means that the graph of a function has no breaks in it. Think of it this way, say you had a graph of a function on a pice of paper. Now suppose you pick up a pencil and follow the graph with your pencil. If you can get from any one point on the graph to anyother point on the graph without having to lift your pencil off the page then the function is continuous.

    Of course maths isn't quite so simple. You didn't think it was did you?? Hehe. Um we can have what are called peicewise continuous functions. Basically thinking of the pencil and graph example again. If you do have to lift your pencil off the page you only have to a finite number of times. In other words the graph is in peices, with breaks between the pieces, but each individual peice is continuous, and there are only a finite number of these peices.

    Discontinuity is basically the regions where a function is not continuous, so looking at peicewise continuous functions another way we could describe them as functions with only finitely many discontinuities. There are a few types of discontinuites, but I'll let someone not studying for exams elaborate on this one. :smile:
  5. May 30, 2003 #4
    Re: What text are you using?

    SFAW - Calculus - Looks like Blue and Red mix.
  6. May 30, 2003 #5
    Interesting tutorials

    This website contains interesting tutorials on caculus, including the concept of continuity in Chapter 1, lesson 5. Hope you enjoy it.
  7. May 30, 2003 #6
    So why are there a bunch of rules and limits about continuity if it's just a property?
  8. May 30, 2003 #7
    I got a little confused on the Intermediate Theorem, what was that a bout?
  9. May 30, 2003 #8
    (please copy and past the above link to your brower)
    (This picture is extracted from http://www.calculus-help.com/funstuff/phobe.html and was modified a bit)

    In the picture, there is a blue curve on the graph. Let this curve be a function f(x) which is continuous on an interval (a,b).

    As you can see, point c is in between a and b on the x-axis. Point d is in between f(a) and f(b) on the y-axis. The blue curve is a continuous one on the interval (a,b).

    The intermediate value theorem tells us if we pick a point, c, arbitrarily in between a and b, f(c) must be in between f(a) and f(b), where f(c) is the point d as shown on the graph.

    The exact defination is on the bottom of the graph. (This is not a difficult concept. Don't be scared by the term "The intermediate theorem")

    This is one of the properties of a continuous function.
  10. May 31, 2003 #9
    this sounded much like that Sandwich Theorem, the lim of a function is between two other functions if they have the same limit.
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