deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 9b9c9dd..d5f09c9 100644  --- a/untitled.tex  +++ b/untitled.tex 
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Then $||X-A||+||B|| ||X-A||+||A|| ||Y-B||$ $$\leq (1+||A||+||B||)\sqrt {(X-A)^2+(Y-B)^2}$$  $$\leq(1+||A||+||B||)\epsilon$$  $$\leq(1+||A||+||B||) \frac {\delta}{1+||A||+||B||}$$ $$=\delta$$  So, given $\epsilon>0$, set $\delta= min(1,\frac {δ}{1+||A||+||B||})$ {\delta}{1+||A||+||B||})$  we have $\sqrt {(X-A)^2+(Y-B)^2}<\delta$  Therefore, for any $(A,B)$ in $R^{2n}$, $g(X, Y) = X · Y$ is continuous. 
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  By Theorem 2.1, every linear function from $R_n$ to $R_m$ can be written as $LA(H)=CH$ for all $H$, where $C$ is some $m × n$ matrix.    (b) The absolute value of each component of a vector is less than or equal to the norm of the vector so for each H and i we hae have  $$0 \leq| f_i(A + H) − f_i(A) − C_i · H| \leq ||F(A + H) − F(A) − CH||$$ where $Ci$ is the $i-th$ row of $C$.   Since $$\frac{||F(A+H)−F(A)=LA(H)||}{||H||}$$ tends to 0,