If f is uniformly continuous on S and has a continuous extension on $overline S$, $tilde f$, then $tilde f$...
Suppose $f : S to mathbb{R}$ is a uniformly continuous function. Then f has a continuous extension $tilde f : overline{S} to mathbb{R}$. Verify that $tilde f$ is continuous on $overline S$.
I am having trouble solving this problem, any hints would be greatly appreciated.
real-analysis continuity
asked Nov 28 at 4:45
Pika_2018
204
add a comment |
Suppose $f : S to mathbb{R}$ is a uniformly continuous function. Then f has a continuous extension $tilde f : overline{S} to mathbb{R}$. Verify that $tilde f$ is continuous on $overline S$.
I am having trouble solving this problem, any hints would be greatly appreciated.
real-analysis continuity
asked Nov 28 at 4:45
Pika_2018
204
3
Hint : how is $bar f$ defined on elements of $bar S setminus S$? Use the fact that a uniformly continuous function carries Cauchy sequences to Cauchy sequences.
– астон вілла олоф мэллбэрг
Nov 28 at 4:48
add a comment |
Suppose $f : S to mathbb{R}$ is a uniformly continuous function. Then f has a continuous extension $tilde f : overline{S} to mathbb{R}$. Verify that $tilde f$ is continuous on $overline S$.
I am having trouble solving this problem, any hints would be greatly appreciated.
real-analysis continuity
asked Nov 28 at 4:45
Pika_2018
204
Suppose $f : S to mathbb{R}$ is a uniformly continuous function. Then f has a continuous extension $tilde f : overline{S} to mathbb{R}$. Verify that $tilde f$ is continuous on $overline S$.
I am having trouble solving this problem, any hints would be greatly appreciated.
real-analysis continuity
real-analysis continuity
asked Nov 28 at 4:45
Pika_2018
204
asked Nov 28 at 4:45
Pika_2018
204
asked Nov 28 at 4:45
Pika_2018
204
asked Nov 28 at 4:45
Pika_2018
204
asked Nov 28 at 4:45
Pika_2018
204
204
3
Hint : how is $bar f$ defined on elements of $bar S setminus S$? Use the fact that a uniformly continuous function carries Cauchy sequences to Cauchy sequences.
– астон вілла олоф мэллбэрг
Nov 28 at 4:48
add a comment |
3
Hint : how is $bar f$ defined on elements of $bar S setminus S$? Use the fact that a uniformly continuous function carries Cauchy sequences to Cauchy sequences.
– астон вілла олоф мэллбэрг
Nov 28 at 4:48
3
3
Hint : how is $bar f$ defined on elements of $bar S setminus S$? Use the fact that a uniformly continuous function carries Cauchy sequences to Cauchy sequences.
– астон вілла олоф мэллбэрг
Nov 28 at 4:48
Hint : how is $bar f$ defined on elements of $bar S setminus S$? Use the fact that a uniformly continuous function carries Cauchy sequences to Cauchy sequences.
– астон вілла олоф мэллбэрг
Nov 28 at 4:48
add a comment |
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Hint : how is $bar f$ defined on elements of $bar S setminus S$? Use the fact that a uniformly continuous function carries Cauchy sequences to Cauchy sequences.
– астон вілла олоф мэллбэрг
Nov 28 at 4:48