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Unformatted text preview: nt of E . If f is di!erentiable at x0 , then f is continuous at x0 .
Proof. Let T be the di!erential of f at x0 . Deﬁne
R (x; x0 ) := f (x) % f (x0 ) % T (x % x0 ) .
Note that by the deﬁnition of di!erentiability
lim x!x0 R (x; x0 )
&x % x0 & 6 Then
f (x) % f (x0 ) = T (x % x0 ) + R (x; x0 ) = b · (x % x0 ) + R (x; x0 ) .
Hence, by Cauchy’s inequality for x # E , x '= x0 ,
0 ( |f (x) % f (x0 )| ( |b · (x % x0 )| + |R (x; x0 )| ( &b& &x % x0 & + |R (x; x0 )|
|R (x; x0 )|
= &x % x0 & &b& +
" 0 (&b& + 0)
&x % x0 &
as x " x0 . It follows by the squeeze theorem that f is continuous at x0 .
Next we study the relation between directional derivatives and di!erentiability. The next theorem gives a formula for the vector b used in the previous
proof and hence determines T . Here we need x0 to be an interior point of E .
Theorem 10 Let E $ RN , let f : E " R be di!erentiable at some point
x0 # E # . Then
(i) all the...
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- Spring '14