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  • - [Voiceover] Which of the following functions

  • are continuous for all real numbers?

  • So let's just remind ourselves

  • what it means to be continuous.

  • What a continuous function looks like.

  • So, a continuous function,

  • let's see, that's my y-axis,

  • that is my x-axis.

  • A function is going to be continuous over some interval.

  • If it just has,

  • doesn't have any jumps or discontinuities

  • over that, or gaps over that interval,

  • so if it's connected

  • and it for sure has to be defined over that interval

  • without any gaps,

  • so for example,

  • a continuous function could look something like this.

  • This function,

  • let me make that line a little bit thicker,

  • so this function right over here is continuous.

  • It is connected over this interval,

  • the interval that we can see.

  • Now, examples of discontinuous functions

  • over an interval, or non-continuous functions,

  • well, they would have gaps of some kind.

  • They could have some type of an asymptotic discontinuity

  • so something like that,

  • that makes it discontinuous.

  • They could have

  • a jump of discontinuity,

  • something like that.

  • They could just have a gap

  • where they're not defined,

  • so they could have a gap where they're not defined,

  • or maybe they actually are defined there,

  • but it's removable discontinuity,

  • so all of these are examples of discontinuous functions.

  • Now, if you want the more mathy understanding of that

  • and we've looked at this before,

  • we say that a function f is continuous,

  • continuous at some

  • value, x equals a,

  • if and only if,

  • draw my little two-way arrows here,

  • say if and only if the limit

  • of f of x as x approaches a

  • is equal to the value of the function at a,

  • so once again, in order to be continuous there,

  • you at least have to be defined there.

  • Now, when you look at these, the one thing that jumps out

  • at me, in order to be continuous for all real numbers,

  • you have to be defined for all real numbers

  • and g of x is not defined for all real numbers.

  • It's not defined for negative values of x,

  • and so, we would

  • rule this one out,

  • so let's think about f of x equals e to the x.

  • It is defined for all real numbers,

  • and as we'll see,

  • most of the common functions that you've learned in math,

  • they don't have these strange jumps or gaps

  • or discontinuities.

  • Some of them do,

  • functions like 1 over x

  • and things like that,

  • but things like e to the x, it doesn't have any of those.

  • We could graph e to the x.

  • E to the x looks something like,

  • e to the x looks something like this,

  • it's defined for all real numbers,

  • there's no jumps or gaps of any kind

  • and so,

  • this f of x is continuous

  • for all real numbers

  • and f only.

  • Now, I didn't do a very rigorous proof.

  • You could if you like,

  • but for the sake of this exercise,

  • it's really more of getting this intuitive sense

  • of like, look, e to the x is defined for all real numbers

  • and so,

  • and there's no jumps or gaps here

  • so it's reasonable to say that it's continuous

  • but you could do a more rigorous proof if you like

  • as well.

- [Voiceover] Which of the following functions

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A2 初級 美國腔

11-2 對所有實數連續的函數(Functions continuous on all real numbers | Limits and continuity | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 05 日
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