Continuous function on $mathbb{Q}$
$begingroup$
Let $f:mathbb{Q}tomathbb{R}$ be a function defined as:
$$f(x) =
begin{cases}
0 & x^2 < 2\
1 & x^2 geq 2
end{cases}
$$ Is this function continuous? How can we check the continuity around $sqrt{2}$ since it's not in $mathbb{Q}$?
calculus real-analysis continuity
$endgroup$
add a comment |
$begingroup$
Let $f:mathbb{Q}tomathbb{R}$ be a function defined as:
$$f(x) =
begin{cases}
0 & x^2 < 2\
1 & x^2 geq 2
end{cases}
$$ Is this function continuous? How can we check the continuity around $sqrt{2}$ since it's not in $mathbb{Q}$?
calculus real-analysis continuity
$endgroup$
$begingroup$
Note (perhaps relevantly) that this function isn't just $mathbb{Q}mapstomathbb{R}$; it's also $mathbb{Q}mapstomathbb{Q}$ (which may be relevant for the next thing you learn using this function!).
$endgroup$
– Steven Stadnicki
Feb 13 '14 at 16:32
add a comment |
$begingroup$
Let $f:mathbb{Q}tomathbb{R}$ be a function defined as:
$$f(x) =
begin{cases}
0 & x^2 < 2\
1 & x^2 geq 2
end{cases}
$$ Is this function continuous? How can we check the continuity around $sqrt{2}$ since it's not in $mathbb{Q}$?
calculus real-analysis continuity
$endgroup$
Let $f:mathbb{Q}tomathbb{R}$ be a function defined as:
$$f(x) =
begin{cases}
0 & x^2 < 2\
1 & x^2 geq 2
end{cases}
$$ Is this function continuous? How can we check the continuity around $sqrt{2}$ since it's not in $mathbb{Q}$?
calculus real-analysis continuity
calculus real-analysis continuity
edited Feb 13 '14 at 8:20
Brian Fitzpatrick
21.1k42958
21.1k42958
asked Feb 13 '14 at 8:16
SpockSpock
950719
950719
$begingroup$
Note (perhaps relevantly) that this function isn't just $mathbb{Q}mapstomathbb{R}$; it's also $mathbb{Q}mapstomathbb{Q}$ (which may be relevant for the next thing you learn using this function!).
$endgroup$
– Steven Stadnicki
Feb 13 '14 at 16:32
add a comment |
$begingroup$
Note (perhaps relevantly) that this function isn't just $mathbb{Q}mapstomathbb{R}$; it's also $mathbb{Q}mapstomathbb{Q}$ (which may be relevant for the next thing you learn using this function!).
$endgroup$
– Steven Stadnicki
Feb 13 '14 at 16:32
$begingroup$
Note (perhaps relevantly) that this function isn't just $mathbb{Q}mapstomathbb{R}$; it's also $mathbb{Q}mapstomathbb{Q}$ (which may be relevant for the next thing you learn using this function!).
$endgroup$
– Steven Stadnicki
Feb 13 '14 at 16:32
$begingroup$
Note (perhaps relevantly) that this function isn't just $mathbb{Q}mapstomathbb{R}$; it's also $mathbb{Q}mapstomathbb{Q}$ (which may be relevant for the next thing you learn using this function!).
$endgroup$
– Steven Stadnicki
Feb 13 '14 at 16:32
add a comment |
5 Answers
5
active
oldest
votes
$begingroup$
Because $sqrt{2}$ is not in $mathbb{Q}$, you don't have to check continuity at $sqrt{2}$: it's completely irrelevant!
$endgroup$
add a comment |
$begingroup$
Hint: You are right, you need only check continuity around (i.e. nearby) $sqrt 2$, not at $sqrt 2$.
$endgroup$
add a comment |
$begingroup$
$f$ is continuous if it is continuous in every point in $mathbb Q$. You don't have to check for continuity around a point that does not exist in the domain.
Hint: for all other points, you can see that there exists a neighborhood on which $f$ is constant.
$endgroup$
add a comment |
$begingroup$
Since the image of $f$ is discrete, we need only check that the fibers of $f$ are open. But
$$
f^{-1}(0)=(-infty,sqrt{2})capmathbb Q
quadtext{and}quad
f^{-1}(1)=(sqrt{2},infty)capmathbb Q
$$
so $f$ is continuous.
$endgroup$
add a comment |
$begingroup$
Try to apply the definition of the continuity. Here's my solution.
Let $epsilon > 0$ and let $x_0 in mathbb{Q}$. Now set $delta$ := |$sqrt{2}-x_0$|. Then, clearly $delta neq 0$ and $delta >0$.
Let $x in mathbb{Q}$ such that |$x-x_0$| < $delta$
then if $x_0 > sqrt{2}$ we have $x > sqrt{2}$ and so $f(x)=1=f(x_0)$
and if $x_0 < sqrt{2}$ we have $x < sqrt{2} $ and so $f(x)=0=f(x_0)$
So we have | $f(x)- f(x_0)$ | $= 0 < epsilon$
Hence $f$ is continuous in $x_0 in mathbb{Q}$ and since this calculation holds for any such $x_0 in mathbb{Q}$, we have that $f$ is continuous on $mathbb{Q}$.
$endgroup$
add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Because $sqrt{2}$ is not in $mathbb{Q}$, you don't have to check continuity at $sqrt{2}$: it's completely irrelevant!
$endgroup$
add a comment |
$begingroup$
Because $sqrt{2}$ is not in $mathbb{Q}$, you don't have to check continuity at $sqrt{2}$: it's completely irrelevant!
$endgroup$
add a comment |
$begingroup$
Because $sqrt{2}$ is not in $mathbb{Q}$, you don't have to check continuity at $sqrt{2}$: it's completely irrelevant!
$endgroup$
Because $sqrt{2}$ is not in $mathbb{Q}$, you don't have to check continuity at $sqrt{2}$: it's completely irrelevant!
answered Feb 13 '14 at 8:20
HurkylHurkyl
111k9118262
111k9118262
add a comment |
add a comment |
$begingroup$
Hint: You are right, you need only check continuity around (i.e. nearby) $sqrt 2$, not at $sqrt 2$.
$endgroup$
add a comment |
$begingroup$
Hint: You are right, you need only check continuity around (i.e. nearby) $sqrt 2$, not at $sqrt 2$.
$endgroup$
add a comment |
$begingroup$
Hint: You are right, you need only check continuity around (i.e. nearby) $sqrt 2$, not at $sqrt 2$.
$endgroup$
Hint: You are right, you need only check continuity around (i.e. nearby) $sqrt 2$, not at $sqrt 2$.
answered Feb 13 '14 at 8:20
Hagen von EitzenHagen von Eitzen
278k22269497
278k22269497
add a comment |
add a comment |
$begingroup$
$f$ is continuous if it is continuous in every point in $mathbb Q$. You don't have to check for continuity around a point that does not exist in the domain.
Hint: for all other points, you can see that there exists a neighborhood on which $f$ is constant.
$endgroup$
add a comment |
$begingroup$
$f$ is continuous if it is continuous in every point in $mathbb Q$. You don't have to check for continuity around a point that does not exist in the domain.
Hint: for all other points, you can see that there exists a neighborhood on which $f$ is constant.
$endgroup$
add a comment |
$begingroup$
$f$ is continuous if it is continuous in every point in $mathbb Q$. You don't have to check for continuity around a point that does not exist in the domain.
Hint: for all other points, you can see that there exists a neighborhood on which $f$ is constant.
$endgroup$
$f$ is continuous if it is continuous in every point in $mathbb Q$. You don't have to check for continuity around a point that does not exist in the domain.
Hint: for all other points, you can see that there exists a neighborhood on which $f$ is constant.
answered Feb 13 '14 at 8:21
5xum5xum
90.2k394161
90.2k394161
add a comment |
add a comment |
$begingroup$
Since the image of $f$ is discrete, we need only check that the fibers of $f$ are open. But
$$
f^{-1}(0)=(-infty,sqrt{2})capmathbb Q
quadtext{and}quad
f^{-1}(1)=(sqrt{2},infty)capmathbb Q
$$
so $f$ is continuous.
$endgroup$
add a comment |
$begingroup$
Since the image of $f$ is discrete, we need only check that the fibers of $f$ are open. But
$$
f^{-1}(0)=(-infty,sqrt{2})capmathbb Q
quadtext{and}quad
f^{-1}(1)=(sqrt{2},infty)capmathbb Q
$$
so $f$ is continuous.
$endgroup$
add a comment |
$begingroup$
Since the image of $f$ is discrete, we need only check that the fibers of $f$ are open. But
$$
f^{-1}(0)=(-infty,sqrt{2})capmathbb Q
quadtext{and}quad
f^{-1}(1)=(sqrt{2},infty)capmathbb Q
$$
so $f$ is continuous.
$endgroup$
Since the image of $f$ is discrete, we need only check that the fibers of $f$ are open. But
$$
f^{-1}(0)=(-infty,sqrt{2})capmathbb Q
quadtext{and}quad
f^{-1}(1)=(sqrt{2},infty)capmathbb Q
$$
so $f$ is continuous.
edited Feb 13 '14 at 16:30
answered Feb 13 '14 at 8:31
Brian FitzpatrickBrian Fitzpatrick
21.1k42958
21.1k42958
add a comment |
add a comment |
$begingroup$
Try to apply the definition of the continuity. Here's my solution.
Let $epsilon > 0$ and let $x_0 in mathbb{Q}$. Now set $delta$ := |$sqrt{2}-x_0$|. Then, clearly $delta neq 0$ and $delta >0$.
Let $x in mathbb{Q}$ such that |$x-x_0$| < $delta$
then if $x_0 > sqrt{2}$ we have $x > sqrt{2}$ and so $f(x)=1=f(x_0)$
and if $x_0 < sqrt{2}$ we have $x < sqrt{2} $ and so $f(x)=0=f(x_0)$
So we have | $f(x)- f(x_0)$ | $= 0 < epsilon$
Hence $f$ is continuous in $x_0 in mathbb{Q}$ and since this calculation holds for any such $x_0 in mathbb{Q}$, we have that $f$ is continuous on $mathbb{Q}$.
$endgroup$
add a comment |
$begingroup$
Try to apply the definition of the continuity. Here's my solution.
Let $epsilon > 0$ and let $x_0 in mathbb{Q}$. Now set $delta$ := |$sqrt{2}-x_0$|. Then, clearly $delta neq 0$ and $delta >0$.
Let $x in mathbb{Q}$ such that |$x-x_0$| < $delta$
then if $x_0 > sqrt{2}$ we have $x > sqrt{2}$ and so $f(x)=1=f(x_0)$
and if $x_0 < sqrt{2}$ we have $x < sqrt{2} $ and so $f(x)=0=f(x_0)$
So we have | $f(x)- f(x_0)$ | $= 0 < epsilon$
Hence $f$ is continuous in $x_0 in mathbb{Q}$ and since this calculation holds for any such $x_0 in mathbb{Q}$, we have that $f$ is continuous on $mathbb{Q}$.
$endgroup$
add a comment |
$begingroup$
Try to apply the definition of the continuity. Here's my solution.
Let $epsilon > 0$ and let $x_0 in mathbb{Q}$. Now set $delta$ := |$sqrt{2}-x_0$|. Then, clearly $delta neq 0$ and $delta >0$.
Let $x in mathbb{Q}$ such that |$x-x_0$| < $delta$
then if $x_0 > sqrt{2}$ we have $x > sqrt{2}$ and so $f(x)=1=f(x_0)$
and if $x_0 < sqrt{2}$ we have $x < sqrt{2} $ and so $f(x)=0=f(x_0)$
So we have | $f(x)- f(x_0)$ | $= 0 < epsilon$
Hence $f$ is continuous in $x_0 in mathbb{Q}$ and since this calculation holds for any such $x_0 in mathbb{Q}$, we have that $f$ is continuous on $mathbb{Q}$.
$endgroup$
Try to apply the definition of the continuity. Here's my solution.
Let $epsilon > 0$ and let $x_0 in mathbb{Q}$. Now set $delta$ := |$sqrt{2}-x_0$|. Then, clearly $delta neq 0$ and $delta >0$.
Let $x in mathbb{Q}$ such that |$x-x_0$| < $delta$
then if $x_0 > sqrt{2}$ we have $x > sqrt{2}$ and so $f(x)=1=f(x_0)$
and if $x_0 < sqrt{2}$ we have $x < sqrt{2} $ and so $f(x)=0=f(x_0)$
So we have | $f(x)- f(x_0)$ | $= 0 < epsilon$
Hence $f$ is continuous in $x_0 in mathbb{Q}$ and since this calculation holds for any such $x_0 in mathbb{Q}$, we have that $f$ is continuous on $mathbb{Q}$.
answered Dec 6 '18 at 13:01
Alain.KlbtrAlain.Klbtr
827
827
add a comment |
add a comment |
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$begingroup$
Note (perhaps relevantly) that this function isn't just $mathbb{Q}mapstomathbb{R}$; it's also $mathbb{Q}mapstomathbb{Q}$ (which may be relevant for the next thing you learn using this function!).
$endgroup$
– Steven Stadnicki
Feb 13 '14 at 16:32