Mixing
Defining various forms of continuity in a (semi)metric space with a $rho$-relaxed triangle inequality.
$begingroup$
Suppose that $(X, d)$ is a semimetric space with a relaxed triangle inequality, i.e., $d: X times X to mathbb{R}$ satisfies for all $x, y, z in X$
begin{align*}
&d(x, y) geq 0 \
&d(x,y) = 0 iff x = y \
&d(x, y) = d(y, x) \
&d(x, z) leq rho (d(x, y) + d(y, z))
end{align*}
I'm wondering how to define various forms of continuity for functionals $f: X to mathbb{R}$ in this space. Regular continuity and uniform continuity are easy and we can define them the usual way:
Continuity:
[
forall x in X: forall varepsilon > 0 : exists delta > 0 : forall y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]Uniform Continuity:
[
forall varepsilon > 0 : exists delta > 0 : forall x,y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]
What about something like Lipschitz continuity? We could certainly use the regular definition without thinking about it and say there exists some $K geq 0$ for all $x, y in X$:
[
|f(x) - f(y)| leq K d(x, y)
]
Or, equivalently, we could write it like our other continuity definitions and say:
[
d(x,y) < frac{varepsilon}{K} implies |f(x) - f(y)| < varepsilon
]
But this got me thinking that since we have the weaker form of the triangle inequality, is there something similar to Lipschitz continuity but involves the constant $rho$ that still remains meaningful?
Basically what I'm asking is, since we have a weaker triangle inequality, can we use the information of $rho$ to define a stronger notion of continuity than just continuity that involves the constant $rho$ that's somewhat meaningful?
continuity metric-spaces
asked Feb 2 at 0:11
RahulRahul
11
$endgroup$
add a comment |
$begingroup$
Suppose that $(X, d)$ is a semimetric space with a relaxed triangle inequality, i.e., $d: X times X to mathbb{R}$ satisfies for all $x, y, z in X$
begin{align*}
&d(x, y) geq 0 \
&d(x,y) = 0 iff x = y \
&d(x, y) = d(y, x) \
&d(x, z) leq rho (d(x, y) + d(y, z))
end{align*}
I'm wondering how to define various forms of continuity for functionals $f: X to mathbb{R}$ in this space. Regular continuity and uniform continuity are easy and we can define them the usual way:
Continuity:
[
forall x in X: forall varepsilon > 0 : exists delta > 0 : forall y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]Uniform Continuity:
[
forall varepsilon > 0 : exists delta > 0 : forall x,y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]
What about something like Lipschitz continuity? We could certainly use the regular definition without thinking about it and say there exists some $K geq 0$ for all $x, y in X$:
[
|f(x) - f(y)| leq K d(x, y)
]
Or, equivalently, we could write it like our other continuity definitions and say:
[
d(x,y) < frac{varepsilon}{K} implies |f(x) - f(y)| < varepsilon
]
But this got me thinking that since we have the weaker form of the triangle inequality, is there something similar to Lipschitz continuity but involves the constant $rho$ that still remains meaningful?
Basically what I'm asking is, since we have a weaker triangle inequality, can we use the information of $rho$ to define a stronger notion of continuity than just continuity that involves the constant $rho$ that's somewhat meaningful?
continuity metric-spaces
asked Feb 2 at 0:11
RahulRahul
11
$endgroup$
add a comment |
$begingroup$
Suppose that $(X, d)$ is a semimetric space with a relaxed triangle inequality, i.e., $d: X times X to mathbb{R}$ satisfies for all $x, y, z in X$
begin{align*}
&d(x, y) geq 0 \
&d(x,y) = 0 iff x = y \
&d(x, y) = d(y, x) \
&d(x, z) leq rho (d(x, y) + d(y, z))
end{align*}
I'm wondering how to define various forms of continuity for functionals $f: X to mathbb{R}$ in this space. Regular continuity and uniform continuity are easy and we can define them the usual way:
Continuity:
[
forall x in X: forall varepsilon > 0 : exists delta > 0 : forall y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]Uniform Continuity:
[
forall varepsilon > 0 : exists delta > 0 : forall x,y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]
What about something like Lipschitz continuity? We could certainly use the regular definition without thinking about it and say there exists some $K geq 0$ for all $x, y in X$:
[
|f(x) - f(y)| leq K d(x, y)
]
Or, equivalently, we could write it like our other continuity definitions and say:
[
d(x,y) < frac{varepsilon}{K} implies |f(x) - f(y)| < varepsilon
]
But this got me thinking that since we have the weaker form of the triangle inequality, is there something similar to Lipschitz continuity but involves the constant $rho$ that still remains meaningful?
Basically what I'm asking is, since we have a weaker triangle inequality, can we use the information of $rho$ to define a stronger notion of continuity than just continuity that involves the constant $rho$ that's somewhat meaningful?
continuity metric-spaces
asked Feb 2 at 0:11
RahulRahul
11
$endgroup$
Suppose that $(X, d)$ is a semimetric space with a relaxed triangle inequality, i.e., $d: X times X to mathbb{R}$ satisfies for all $x, y, z in X$
begin{align*}
&d(x, y) geq 0 \
&d(x,y) = 0 iff x = y \
&d(x, y) = d(y, x) \
&d(x, z) leq rho (d(x, y) + d(y, z))
end{align*}
I'm wondering how to define various forms of continuity for functionals $f: X to mathbb{R}$ in this space. Regular continuity and uniform continuity are easy and we can define them the usual way:
Continuity:
[
forall x in X: forall varepsilon > 0 : exists delta > 0 : forall y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]Uniform Continuity:
[
forall varepsilon > 0 : exists delta > 0 : forall x,y in X qquad d(x, y) < delta implies |f(x) - f(y)| < varepsilon
]
What about something like Lipschitz continuity? We could certainly use the regular definition without thinking about it and say there exists some $K geq 0$ for all $x, y in X$:
[
|f(x) - f(y)| leq K d(x, y)
]
Or, equivalently, we could write it like our other continuity definitions and say:
[
d(x,y) < frac{varepsilon}{K} implies |f(x) - f(y)| < varepsilon
]
But this got me thinking that since we have the weaker form of the triangle inequality, is there something similar to Lipschitz continuity but involves the constant $rho$ that still remains meaningful?
Basically what I'm asking is, since we have a weaker triangle inequality, can we use the information of $rho$ to define a stronger notion of continuity than just continuity that involves the constant $rho$ that's somewhat meaningful?
continuity metric-spaces
continuity metric-spaces
asked Feb 2 at 0:11
RahulRahul
11
asked Feb 2 at 0:11
RahulRahul
11
asked Feb 2 at 0:11
RahulRahul
11
asked Feb 2 at 0:11
RahulRahul
11
asked Feb 2 at 0:11
RahulRahul
11
11
add a comment |
add a comment |
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