From Vectorcalcumb

Vector Extreme Value Theorem

<m>F: \\bbR^n \\right \\bbR^m</m>
<m>F</m> is continuous, <m>A \\subset \\bbR^n</m>
<m>\\right</m> F is bounded and obtains miminum and maximum values
i.e. <m> F(A)</m> is compact

And apparently, there's no analogue to the Intermediate Value Theorem in Vector Calculus.

And then I have these scribblings:

<m> F(A)\\subset \\bigcup{i \\in I}{}{u_i} </m>

<m> A \\subset \\bigcup{i \\in I}{}{F^{-1}u_i} </m>

<m> A \\subset \\bigcup{k=1}{n}{F^{-1}u_i_k} \\doubleright F(A) \\subset \\bigcup{k=1}{n}{u_i_k}</m>