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Unformatted text preview: ∈ R, then the limit of its
restriction on any curve would also be . However, here we found at least two curves
— straight line = 0 and parabola = 2 such that the respective restrictions have
diﬀerent limits. Therefore lim( )→(0 0) (
) is not deﬁned.
:)
5 E Consider the function ( 2
2 )= 2
2 + 0 2 +
+ 2 2 =0
=0 Prove that is continuous with respect to for each ﬁxed = , continuous with respect
to for each ﬁxed = , but discontinuous at the point (0 0).
S For each ﬁxed = , if = 0, then by deﬁnition,
2
( )= 2 2 + It’s easy to see that (
) is a rational function of
with nonzero denominator and
therefore we know it’s continuous with respect to from calculus of one variable. If
= 0, then it’s easy to see from the deﬁnition that ( 0) = 0 so it’s also continuous.
Similarly, for each ﬁxed = , if = 0, then by deﬁnition,
( 2
2+ )= 2 It’s easy to see that (
) is a rational function of with nonzero denominator and
therefore we know it’s continuous with respect to from calculus of one variable. I...
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This document was uploaded on 02/10/2014.
 Spring '13

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