Are continuous mappings on a compact metric space Lipschitz?
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Continuous mapping on a compact metric space is uniformly continuous is a standard result in real analysis. Lipschitz functions are uniformly continuous. Can the aforementioned result be generalized to Lipschitz? i.e. are all continuous functions on compact metric spaces Lipschitz?
Can we require anything more so that all continuous functions are Lipschitz?
real-analysis metric-spaces
asked Nov 15 at 15:19
Baran Zadeoglu
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up vote
0
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Continuous mapping on a compact metric space is uniformly continuous is a standard result in real analysis. Lipschitz functions are uniformly continuous. Can the aforementioned result be generalized to Lipschitz? i.e. are all continuous functions on compact metric spaces Lipschitz?
Can we require anything more so that all continuous functions are Lipschitz?
real-analysis metric-spaces
asked Nov 15 at 15:19
Baran Zadeoglu
276
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Continuous mapping on a compact metric space is uniformly continuous is a standard result in real analysis. Lipschitz functions are uniformly continuous. Can the aforementioned result be generalized to Lipschitz? i.e. are all continuous functions on compact metric spaces Lipschitz?
Can we require anything more so that all continuous functions are Lipschitz?
real-analysis metric-spaces
asked Nov 15 at 15:19
Baran Zadeoglu
276
Continuous mapping on a compact metric space is uniformly continuous is a standard result in real analysis. Lipschitz functions are uniformly continuous. Can the aforementioned result be generalized to Lipschitz? i.e. are all continuous functions on compact metric spaces Lipschitz?
Can we require anything more so that all continuous functions are Lipschitz?
real-analysis metric-spaces
real-analysis metric-spaces
asked Nov 15 at 15:19
Baran Zadeoglu
276
asked Nov 15 at 15:19
Baran Zadeoglu
276
asked Nov 15 at 15:19
Baran Zadeoglu
276
asked Nov 15 at 15:19
Baran Zadeoglu
276
asked Nov 15 at 15:19
Baran Zadeoglu
276
276
add a comment |
add a comment |
2 Answers
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First question: No. For example, $f(x) = sqrt{x}$ on the compact interval $[0,1]$ isn't Lipschitz.
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
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1
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See $sqrt X $on [0,1] is satisfies given condition but not Lipschitz
answered Nov 15 at 15:27
Shubham
1,2581518
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
First question: No. For example, $f(x) = sqrt{x}$ on the compact interval $[0,1]$ isn't Lipschitz.
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
add a comment |
up vote
2
down vote
accepted
First question: No. For example, $f(x) = sqrt{x}$ on the compact interval $[0,1]$ isn't Lipschitz.
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
First question: No. For example, $f(x) = sqrt{x}$ on the compact interval $[0,1]$ isn't Lipschitz.
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
First question: No. For example, $f(x) = sqrt{x}$ on the compact interval $[0,1]$ isn't Lipschitz.
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
answered Nov 15 at 15:24
Hans Lundmark
34.5k564110
34.5k564110
add a comment |
add a comment |
up vote
1
down vote
See $sqrt X $on [0,1] is satisfies given condition but not Lipschitz
answered Nov 15 at 15:27
Shubham
1,2581518
add a comment |
up vote
1
down vote
See $sqrt X $on [0,1] is satisfies given condition but not Lipschitz
answered Nov 15 at 15:27
Shubham
1,2581518
add a comment |
up vote
1
down vote
up vote
1
down vote
See $sqrt X $on [0,1] is satisfies given condition but not Lipschitz
answered Nov 15 at 15:27
Shubham
1,2581518
See $sqrt X $on [0,1] is satisfies given condition but not Lipschitz
answered Nov 15 at 15:27
Shubham
1,2581518
answered Nov 15 at 15:27
Shubham
1,2581518
answered Nov 15 at 15:27
Shubham
1,2581518
answered Nov 15 at 15:27
Shubham
1,2581518
1,2581518
add a comment |
add a comment |
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