2.Limit and Continuity of Functions with Several Variables

Limit

Definition

Let $f:R^n\mapsto R$ where $D=dom(f)\subseteq R^n$, $p\in D$. We say that limit of f at $p_0$ is $L\in R$. We say that limit of $f$ at $p_0$ is $L\in R$ and we write

$li{m}_{p\to p_0}f(p)=L$

if $\forall ϵ>0,\exists \sigma >0$ st. $0<|p-p_0|<\sigma ⟹|f(p)-L|<ϵ$.

Remember that if $h(X)$ is cont. then $li{m}_{p\to p_0}h(p)=h(li{m}_{p\to p_0})$

Warning

L’H is NOT applicable for two variable functions.

Todo

• Examples and exercises in Week 3 Note 2

Continuity

Definition

$z=f(x,y)$ is continuous at $p(x_0,y_0)⟺li{m}_{(x,y)\to (x_0,y_0)}f(x,y)=f(x_0,y_0)$

Note that $z=f(x,y)$ is continuous at $D\subseteq R^2$ if $f(x,y)$ is continuous at each point of $D$.

Theorem

Let $f:R^n\to R^m$ be continuous on $R^n$. Then, for all open/closed subset $S\subseteq R^n$, $f^{-1}(S)$ is also open/closed.

Take $S=\{(x,y)\in R^2|si{n}^2(xy)\le 1\text{ and }{x}^2+y^2\le 1\}$. Observe that $S=(si{n}^2(x,y){)}^{-1}([0,1])\cap (x^2+y^2{)}^{-1}([0,1])$. Both functions are continuous on $R$ and the sets are closed. Then $S$ is closed. Important part here is that first you should check if the function is continuous on $R^n$.

Uniform Continuity

Theorem

If $f:D\to R$ is continuous and $D$ is compact, then $f$ is uniformly continuous on $D$.

Remark

If $f:R\to R$ and $|f^{\prime }(x)|\le K$, $\forall x\in R$, then $f$ is uniformly continuous on $R$.