Continuity

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We investigate what continuity means for functions of several variables.

Now that we have defined limits, we can define continuity.

Let $F:\R ^n\to \R$ and $\pt {a}$ be an interior point of the domain of $F$ . We say $F$ is continuous at $\pt {x}=\pt {a}$ , if

$F(\pt {a})$ exists.
$\lim _{\pt {x}\to \pt {a}} F(\pt {x})$ exists.
$\lim _{\pt {x}\to \pt {a}} F(\pt {x}) = F(\pt {a}).$

$F$ is continuous on an open set $S$ if $F$ is continuous at all $\pt {x} \in S$ .

The limit laws can be used to write corresponding continuity laws.

Limit Laws Let $F:\R ^n\to \R$ and $G:\R ^n\to \R$ be continuous functions of several variables, and $b$ be a real number. $\begin{align*} \pt{x} &= \point{x_1,x_2,\dots,x_n}\\ \pt{a} &= \point{a_1,a_2,\dots,a_n}, \end{align*}$

where $\lim _{\pt {x}\to \pt {a}}F(\pt {x}) = L \quad \text {and}\quad \lim _{\pt {x}\to \pt {a}} G(\pt {x}) = M.$

Constant Law: $F(\pt {x}) = b$ is continuous.
Identity Law: $F(\pt {x}) = x_i$ is continuous.
Sum/Difference Law: $F(\pt {x})\pm G(\pt {x})$ is continuous.
Scalar Multiple Law: $b\cdot F(\pt {x})$ is continuous.
Product Law: $F(\pt {x})\cdot G(\pt {x})$ is continuous.
Quotient Law: $\frac {F(\pt {x})}{G(\pt {x})}$ is continuous where $G(\pt {x}) \neq 0$ .

True or false: If $F:\R ^2\to \R$ and $G:\R ^2\to \R$ are continuous functions on an open disk $B$ , then $F\pm G$ is continuous on $B$ .

True False

True or false: If $F:\R ^2\to \R$ and $G:\R ^2\to \R$ are continuous functions on an open disk $B$ , then $F/G$ is continuous on $B$ .

True False

The function $F/G$ may or may not be continuous, it depends on whether $G(x,y)=0$ . If $G(x,y)=0$ , then $F/G$ not continuous at that point.

Composition Limit Law Let $f:\R \to \R$ be a continuous function on an interval $I$ . Let $G:\R ^n\to \R$ be a function whose range is contained in (or equal to) $I$ , Then $\lim _{\pt {x}\to \pt {a}} f( G(\pt {x})) = f(\lim _{\pt {x}\to \pt {a}}G(\pt {x}))$

Composition of Composite Functions Let $G:\R ^n\to \R$ be continuous on an open disk $B$ , where the range of $G$ on $B$ is $I$ , and let $f$ be a single variable function that is continuous on $I$ . Then $f\circ G(\pt {x}) =f(G(\pt {x})),$ is continuous on $B$ .

Show that the function $F(x,y) = \sin (x^2\cos (y))$ is continuous for all points in $\R ^2$ .

Let $F_1(x,y) = x^2.$ Since $y$ is not actually used in the function, and polynomials are continuousare not continuous , we conclude $F_1$ is continuous everywhere. A similar statement can be made about $F_2(x,y) = \cos (y).$ Setting $F_3=F_1\cdot F_2$ we obtain a continuous function from $\R ^2\to \R$ . Since sine is continuousis not continuous for all real values, the composition of sine with $F_3$ is continuous. Hence, $\sin (F_3(x,y)) = \sin (x^2\cos y)$ is continuous everywhere. We finish by presenting you with a plot of $F$ :

Let $F(x,y) = \begin {cases} \frac {\cos (y)\sin (x)}{x} & x\neq 0 \\ \cos (y) & x=0 \end {cases}$ Is $F$ continuous at $(0,0)$ ? Is $F$ continuous everywhere?

To determine if $F$ is continuous at $(0,0)$ , we need to compare $\lim _{\point {x,y}\to \point {0,0}} F(x,y)\quad \text {to}\quad F(0,0).$ Applying the definition of $F$ , we see that: $F(0,0) = \answer [given]{1}$ We now consider the limit $\lim _{\point {x,y}\to \point {0,0}}F(x,y).$ Substituting $0$ for $x$ and $y$ in $(\cos (y)\sin (x))/x$ returns the indeterminate form $\relax \boldsymbol {\tfrac {0}{0}}$ , so we need to do more work to evaluate this limit.

Consider two related limits:

$\begin{align*} \lim_{\point{x,y}\to\point{0,0}} \cos(y)\\ \lim_{\point{x,y}\to\point{0,0}} \frac{\sin(x)}{x}. \end{align*}$

The first limit does not contain $x$ , and since $\cos (y)$ is continuous,

$\begin{align*} \lim_{\point{x,y}\to\point{0,0}} \cos(y) &=\lim_{y\to 0} \cos(y) \\ &=\answer[given]{1} \end{align*}$

The second limit does not contain $y$ . But we know

$\begin{align*} \lim_{\point{x,y}\to\point{0,0}} \frac{\sin(x)}{x} &= \lim_{x\to 0} \frac{\sin(x)}{x} \\ &= \answer[given]{1}. \end{align*}$

Finally, we know that we can combine these two limits so that $\lim _{\point {x,y}\to \point {0,0}} \frac {\cos (y)\sin (x)}{x}$

$\begin{align*} &= \lim_{\point{x,y}\to\point{0,0}} (\cos(y))\left(\frac{\sin(x)}{x}\right) \\ &=\left(\lim_{\point{x,y}\to\point{0,0}} \cos(y)\right)\left(\lim_{\point{x,y}\to\point{0,0}} \frac{\sin(x)}{x}\right) \\ &=\answer[given]{1}\cdot \answer[given]{1}. \end{align*}$

We have found that $\lim _{\point {x,y}\to \point {0,0}} \frac {\cos (y)\sin (x)}{x} = F(0,0)$ , so $F$ is continuous at $(0,0)$ .

A similar analysis shows that $F$ is continuous at all points in $\mathbb {R}^2$ . As long as $x\neq 0$ , we can evaluate the limit directly; when $x=0$ , a similar analysis shows that the limit is $\cos y$ . Thus we can say that $F$ is continuous everywhere. We finish by presenting you with a plot of $F$ :