4. In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm is differentiable at x = RN then F is continuous at x. Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want F(x), which means F is continuous at x. to show that F(xn) Denote hn xnx, so that ||hn||| 0. Thus we find ||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)|| (**) ||DF(x)hn|| + ||R(hn) || → 0, because the linear mapping DF(x) is continuous and for all large n = N, |||R(hn) || ≤ (***) ||R(hn)|| ||hn|| → 0. Explain the steps labelled (*), (**), (***) [6 Marks] (ii) Give an example of a function F: RR such that F is contin- Total marks 10 uous at x=0 but F is not differentiable at at x = 0. [4 Marks]
4. In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm is differentiable at x = RN then F is continuous at x. Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want F(x), which means F is continuous at x. to show that F(xn) Denote hn xnx, so that ||hn||| 0. Thus we find ||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)|| (**) ||DF(x)hn|| + ||R(hn) || → 0, because the linear mapping DF(x) is continuous and for all large n = N, |||R(hn) || ≤ (***) ||R(hn)|| ||hn|| → 0. Explain the steps labelled (*), (**), (***) [6 Marks] (ii) Give an example of a function F: RR such that F is contin- Total marks 10 uous at x=0 but F is not differentiable at at x = 0. [4 Marks]
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
Related questions
Question
![4.
In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm
is differentiable at x = RN then F is continuous at x.
Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want
F(x), which means F is continuous at x.
to show that F(xn)
Denote hn
xnx, so that ||hn||| 0. Thus we find
||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)||
(**)
||DF(x)hn|| + ||R(hn) || → 0,
because the linear mapping DF(x) is continuous and for all large n = N,
|||R(hn) || ≤
(***) ||R(hn)||
||hn||
→ 0.
Explain the steps labelled (*), (**), (***)
[6 Marks]
(ii)
Give an example of a function F: RR such that F is contin-
Total marks 10
uous at x=0 but F is not differentiable at at x = 0.
[4 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c7573-6777-48ff-9bfb-9b3df1a769a6%2F7aa3b499-daeb-4b7c-91ae-09aff2ccf90c%2F3co3fy6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.
In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm
is differentiable at x = RN then F is continuous at x.
Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want
F(x), which means F is continuous at x.
to show that F(xn)
Denote hn
xnx, so that ||hn||| 0. Thus we find
||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)||
(**)
||DF(x)hn|| + ||R(hn) || → 0,
because the linear mapping DF(x) is continuous and for all large n = N,
|||R(hn) || ≤
(***) ||R(hn)||
||hn||
→ 0.
Explain the steps labelled (*), (**), (***)
[6 Marks]
(ii)
Give an example of a function F: RR such that F is contin-
Total marks 10
uous at x=0 but F is not differentiable at at x = 0.
[4 Marks]
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