must be positive on a neighborhood of that point.) Can you state a somewhat stronger result?6. Let U C R" be open. Suppose g: U R is continuous and g(a) 0. Prove that 1/g is contin-us on some neighborhood of a. (Hint: Apply Proposition 3.5.)7. Suppose T: R"-R" is a linear map.

Question
Asked Aug 16, 2019

I need help for problem 6. Thank you very much!

must be positive on a neighborhood of that point.) Can you state a somewhat stronger result?
6. Let U C R" be open. Suppose g: U R is continuous and g(a) 0. Prove that 1/g is contin-
us on some neighborhood of a. (Hint: Apply Proposition 3.5.)
7. Suppose T: R"-R" is a linear map.
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must be positive on a neighborhood of that point.) Can you state a somewhat stronger result? 6. Let U C R" be open. Suppose g: U R is continuous and g(a) 0. Prove that 1/g is contin- us on some neighborhood of a. (Hint: Apply Proposition 3.5.) 7. Suppose T: R"-R" is a linear map.

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Expert Answer

Step 1

To prove:

1
is continuous on some neighborhood of a
g
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1 is continuous on some neighborhood of a g

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Step 2

Proof:

...
Let UER" be open. Suppose the function g:U->Rbe continuous and g (a)# 0.Then,
g(x) is continuous on every xeU. For some x# a, g(x)=0. Thus, the reciprocal of
1
g can be defined as
1
:g(x) is continuous and g (x)# o^ such that the function
g(x)
1
is continuous on U\x: g(x)=0. Therefore, the function
g
1
is continuous on every
xEU, but not on x= a where g(a)=0
help_outline

Image Transcriptionclose

Let UER" be open. Suppose the function g:U->Rbe continuous and g (a)# 0.Then, g(x) is continuous on every xeU. For some x# a, g(x)=0. Thus, the reciprocal of 1 g can be defined as 1 :g(x) is continuous and g (x)# o^ such that the function g(x) 1 is continuous on U\x: g(x)=0. Therefore, the function g 1 is continuous on every xEU, but not on x= a where g(a)=0

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