21-356 Lecture 3

Proof let t be the dierential of f at x0 dene r x x0

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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 ) = 0. &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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