Name: 什麼是光滑函數？ (What is a smooth function?)
Uploaded: 2025-11-10T15:16:46.000Z
Duration: 2 min 12 s
Description: 【看影片學英語】數萬部 YouTube 影片，搭配英漢字典即點即查，輕鬆掌握單字發音與用法，長久累積看電影不必再看字幕。學這些英文用法：函數,連續,常見,訂閱,影片,例子,等於,幫助,數學,愉快,順利,談談,看到,聊聊,討論,解釋,表現,製作,意義,代表,來到,繼續,找到,看看

to 不定詞

Today we're going to be talking about what does it mean for a function to be smooth?

今天我們要聊聊，一個函數光滑是什麼意思呢？

動名詞

Now, before we get started, if you haven't subscribed, please consider doing so, because it really helps me to be able to keep making these videos.

在我們開始之前，如果你還沒訂閱，請考慮一下，這對我繼續製作這些影片真的很有幫助。

Now, with that very smooth transition out of the way, let's talk about what it means for a function to be smooth.

好了，非常順暢地過渡了這個部分，讓我們來談談函數光滑的意思。

Well, a function is smooth if it has derivatives of every possible order.

嗯，一個函數光滑，如果它擁有所有可能的階數的導數。

So a function is of order k is if the k th derivative of f exists and is continuous.

所以一個函數是 k 階的，如果 f 的 k 階導數存在且連續。

So a function is smooth if it is infinitely differentiable.

所以一個函數光滑，如果它是無窮次可微的。

We can also say that a function f is of order k by saying that F is in C to the k.

我們也可以說函數 f 是 k 階的，說 f 是 C^k。

This is just a common notation that you might see if you are trying to talk about a whole bunch of different orders at once.

這只是一個常見的表示法，如果你想一次討論很多不同的階數時可能會看到。

So by convention, C0 denotes all continuous function, C1 is all functions whose derivative is continuous, C2 is all function whose second derivative is continuous, and

所以依慣例，C0 代表所有連續函數，C1 是所有導數連續的函數，C2 是所有二階導數連續的函數，而

So a function is smooth if it is in c infinity.

所以一個函數光滑，如果它是 C^∞。

So if a function is in ck, then we can also say it's in CJ if j is less than k.

所以如果一個函數是 C^k，那麼它也是 C^j，如果 j 小於 k。

Some common smooth functions are polynomials e to the x cosine of x sine of x1 over x squared plus 1.

一些常見的光滑函數有多項式、e^x、cos(x)、sin(x)、1/(x^2 + 1)。

There's a whole bunch of other ones, but these are some common ones.

還有很多其他的，但這些是一些常見的。

You might see the function absolute value of x that's c0, but

你可能會看到函數 |x|，它是 C0，但是

it's not c1 since the derivative does not exist at x equals 0.

它不是 C1，因為導數在 x 等於 0 時不存在。

So there's an example to show that c0 is not the same thing as c1.

所以這是一個例子，說明 C0 和 C1 並不相同。

And then the function x squared sine of 1 over x that turns out to be c1 but not c2.

然後是函數 x^2 sin(1/x)，它會是 C1 但不是 C2。

And you can find examples of all sorts of different functions that do similar things with the other different orders.

你也可以找到各種不同的函數例子，它們在其他不同的階數上也有類似的表現。

I hope this video was helpful in understanding both the order of derivatives and also what it means for a function to be smooth.

希望這個影片能幫助你理解導數的階數，以及函數光滑的意義。

Please check out some of my other videos and like and subscribe if you haven't already.

如果你還沒訂閱，請看看我其他的影片，並且按讚訂閱喔。

And if there's a different topic you'd like to see me explain in a way similar to this, please leave it in the comment section down below.

如果你有其他想看我用類似這種方式解釋的主題，請在下方的留言區告訴我。

I hope you all have a great rest of your day and good luck with all of your math.

希望大家今天都過得愉快，數學順利！