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    what does "f(x) is continuous" mean in "sign change and f(x) continuous imply that the root lies in the interval"?
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    If the function is continuous, it means that it is basically a graph with a single curve that isn't broken up into other smaller ones such as one that doesn't have asymptotes, a hole in the middle of the curve, etc. It's not quite the mathematical explanation but that's basically what they mean by it. An example of one would be this.

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    (Original post by drinktheoceans)
    If the function is continuous, it means that it is basically a graph with a single curve that isn't broken up into other smaller ones such as one that doesn't have asymptotes, a hole in the middle of the curve, etc. It's not quite the mathematical explanation but that's basically what they mean by it. An example of one would be this.

    so in this graph, the root will be lie in the interval [0,0.5]?
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    (Original post by alesha98)
    what does "f(x) is continuous" mean in "sign change and f(x) continuous imply that the root lies in the interval"?
    Being continuous means, intuitively that you can draw the curve without taking your pen off the paper.

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    Rigorously, it means that at all points, the function equals it's limit at that point. This requires being (a) defined at all points, (b) having a limit defined at all points and (c) those things being equal everywhere. Limits in turn are defined by an ardous process involving deltas and epsilons.


    Since you need to be able to draw the curve, if it goes from positive (above the x-axis) to negative (below the axis) then it MUST have passed through the x-axis on the way. See "We're going on a bear hunt" for the logic involved A root is simply a point where the curve is touching the axis (or crossing).

    Note that continuity is actually important - if you don't take it into account you'd think that 1/x has to have a root in the interval [-1,1] but we know that it doesn't. This is because it is discontinuous.
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    (Original post by lerjj)
    Being continuous means, intuitively that you can draw the curve without taking your pen off the paper.
    Spoiler:
    Show
    Rigorously, it means that at all points, the function equals it's limit at that point. This requires being (a) defined at all points, (b) having a limit defined at all points and (c) those things being equal everywhere. Limits in turn are defined by an ardous process involving deltas and epsilons.

    Since you need to be able to draw the curve, if it goes from positive (above the x-axis) to negative (below the axis) then it MUST have passed through the x-axis on the way. See "We're going on a bear hunt" for the logic involved A root is simply a point where the curve is touching the axis (or crossing).

    Note that continuity is actually important - if you don't take it into account you'd think that 1/x has to have a root in the interval [-1,1] but we know that it doesn't. This is because it is discontinuous.
    ohhhh thankyou so much
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    (Original post by alesha98)
    what does "f(x) is continuous" mean in "sign change and f(x) continuous imply that the root lies in the interval"?
    Take a look at the "Intermediate Value Theorem".
 
 
 
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