Limit and Continuity

Let $D \subset \mathbb{R} ^{n}$, f be a function on D, $c \in \mathbb{R} ^{n}$, and l a fixed number. We say that the limit of f(x) is l when $x \rightarrow c$ if f(x) gets arbitrarily close to l whenever x, while remaining distinct from c, gets sufficiently close c .

這是極限觀念的極粗略的描述, 在下編中我們會有更嚴謹的討論, 和單變數時類似, 若有正數 K 使 c 點附近和 c 相異的點 x 都能滿足 $\vert f(x)-l\vert \leq K \parallel x-c \parallel$, 則定義中的條件顯然符合. 於是 $$\lim_{x \rightarrow c} f(x) = l$$, 這充分條件是證明一些簡單的極限問題時常用的方法.

利用極限概念可定義連續概念如次:
Loosely speaking, a function is said to be continuous at point p if for those values of x where f(x) is defined, $$\lim_{x\rightarrow p} f(x) = f(p)$$. f is said to be continuous on the domain D if it is continuous at every point $x \in D$.

若 $f,g: D\rightarrow \mathbb{R}$ 在 p 點連續, 則 f(x)g(x) 及 af(x)+bg(x), $a,b \in \mathbb{R}$, 也都在 p 點連續. 至於 $\frac{f(x)}{g(x)}$, 其連續的區域可能縮小, 因為我們可能有將使 g(x)=0 的點排的需要 (見例 3)

例 1   設 $f(x_{1},\ldots,x_{n})$ 是 $x_{1},\ldots,x_{n}$ 的多項式, 則 f 處處連續.

例 2   Let

$$f(x,y)=\left\{ \begin{array}{ll} \displaystyle\frac{x^{2} y}{... ...{\smallskip } 0 & \hbox{if } (x,y) = (0,0) \end{array}\right.$$

Is f continuous at (0,0)?

Solution. For $(x.y)\neq (0,0)$ we have

$$0 \leq \vert f(x, y)-f(0,0) \vert=\vert \frac{x^{2} y}{x^{2} ... ...xy}{x^{2} + y^{2} } \, \cdot \, \vert x\vert \leq \vert x\vert$$

Therefore

$$\lim_{(x, y)\rightarrow (0,0)} f(x, y)=f(0,0)$$

f is thus seen to be continuous at (0,0).

例 3   Let

$$f(x,y)=\left\{ \begin{array}{ll} \displaystyle\frac{x^{2} -y^... ... \noalign{\smallskip } 0 & (x,y) = (0,0) \end{array} \right.$$

Is f continuous at (0,0) ?

解. 欲解本問題, 我們考慮 f 在通過原點 (0,0) 的諸直線上的 behavior. 令 $\theta$ 為定角, 滿足 $cos\theta \neq 0$ and $sin\theta\neq 0$, 則在直線

$$x=t\cos \theta \, , y=t\sin \theta \, , \hbox to 1truecm{\hfill} t \in \mathbb{R}$$

上, 下式成立:

$$f(x,y)=f(t\cos \theta , t\sin \theta )=\cos 2\theta, \hbox to 1truecm{\hfill} t\neq 0$$

故知 $lim_{t \rightarrow 0} f(t\cos \theta ,t\sin \theta )=\cos 2\theta$. 因為此值與所論直線織選擇有關, 所以當 $(x,y) \longrightarrow (0,0)$ 時, 所論之函數沒有極限, 故微在原點 (0,0) 處不連續函數.

本例題告訴我們, 若變數沿不同的直線趨於一點時, 函數有不同的極限, 則當變數 (不限制路線) 趨於該點時, 極限不存在. 這例題的逆命題不成立. 讀者可參考下面的第四題. 本例題的另一個有趣的結果是

$$\lim_{x \rightarrow 0} \lim_{y \rightarrow 0} f(x,y)=1 \neq -1=\lim_{x \rightarrow 0} \lim_{y \rightarrow 0} f(x,y)$$

習$\quad$題

1.

求下列二函數的等高線:
(a) $$z=x^{2}+y^{2}$$,      (b) $$z= \frac{1}{x^{2}+y^{2} }$$.

2.

描述下列函數的等位面:
(a) $$u=x+y+z$$,     (b) $$u=x^{2}+y^{2}+z^{2}$$.

3.

計算下列極限之值: (a) $$\lim_{(x,y) \rightarrow (0,0)} x^{2}y^{2} \ln(x^{2} +y^{2} )$$,(b) $$\lim_{(x,y) \rightarrow (0,0)} (x^{2} +y^{2} )^{x^{2} y^{2} }$$.

4.

Consider the function $z= \frac{2xy^{2} }{x^{2} +y^{4} }$. Show that z approaches 0when (x,y) moves along any straight line toward the origin while z approaches different limits when (x,y) moves along the two parabolas $x= \pm y^{2}$.

5.

設 $$f(x,y)=\exp(\frac{x}{x^{2} +y^{2}})$$. 問 $\theta$ 為何值時 $$\lim_{t \longrightarrow 0} f(t\cos \theta ,t\sin \theta )$$ 存在?

6.

Sketch the level curves for the surface whose equation is

z=1+(x² +y² ) e^{-(x2 +y2 )}.

Determine the maximum on this surface.

7.

Determium the valuds of the following functions f(x,y)

$$f(x,y)= \left\{ \begin{array}{ll} \displaystyle\frac{xy}{x^{2... ...\\ \noalign{\smallskip } 0, & (x,y)=(0,0). \end{array}\right.$$

along the lines y=mx in terms of x. Are they continuous at the origin?

8.

令

$$f(x,y)= \left\{ \begin{array}{ll} \displaystyle\frac{y^{3} }{... ...{\smallskip } 0, & \mbox{當 } (x,y)=(0,0). \end{array}\right.$$

(a) 問 $$\lim_{x \rightarrow 0} \lim_{y \longrightarrow 0} f(x,y)f(x,y)$$是否存在? 為什麼?
(b) f 在原點是否連續?