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Is anyone familiar with any resources on the study of continuity of functions on the real line via gauges?
This is inspired by the gauge integral. Briefly, a gauge on a closed and bounded interval [tex]I \subseteq \mathbb{R}[/tex] is a strictly positive function [tex]\delta : I \rightarrow \mathbb{R}[/tex]. Let [tex]\{(I_i,t_i)\}_{i=1}^n[/tex] be a tagged partition of [tex]I[/tex]; this partition is [tex]\delta[/tex]-fine if [tex]I_i \subset [t_i - \delta(t_i), t_i + \delta(t_i)][/tex]; the existence of [tex]\delta[/tex]-fine partitions for any gauge [tex]\delta[/tex] can be shown from the completeness of [tex]\mathbb{R}[/tex].
Continuity of [tex]f[/tex] on [tex]I = [a,b][/tex] induces a natural gauge for each [tex]\epsilon[/tex] (fix a point [tex]t \in I[/tex] and an [tex]\epsilon>0[/tex]; then the [tex]\delta[/tex] corresponding to [tex]x \in I[/tex] is defined to be the value of the gauge at [tex]x[/tex]). From this particular gauge many standard theorems can be proved, e.g., the intermediate value theorem or the fact that a continuous function on a closed, bounded interval is uniformly continuous. Most remarkable is how short the proofs are. Many are nearly immediate, at least compared to the standard proofs I've been exposed to before.
I've found this approach really interesting and have been looking for more resources on it. I wonder if continuity can be defined in terms of gauges. And I wonder how well this generalizes (say to metric or topological spaces).
Anyone know about this? Thanks!
Edit:
I'm wanting to find as many resources on this since it seems to be a relatively unexplored topic and am planning to author an expository article discussing it, unless to my surprise I find many such articles which heretofore appear to be nonexistent.
This is inspired by the gauge integral. Briefly, a gauge on a closed and bounded interval [tex]I \subseteq \mathbb{R}[/tex] is a strictly positive function [tex]\delta : I \rightarrow \mathbb{R}[/tex]. Let [tex]\{(I_i,t_i)\}_{i=1}^n[/tex] be a tagged partition of [tex]I[/tex]; this partition is [tex]\delta[/tex]-fine if [tex]I_i \subset [t_i - \delta(t_i), t_i + \delta(t_i)][/tex]; the existence of [tex]\delta[/tex]-fine partitions for any gauge [tex]\delta[/tex] can be shown from the completeness of [tex]\mathbb{R}[/tex].
Continuity of [tex]f[/tex] on [tex]I = [a,b][/tex] induces a natural gauge for each [tex]\epsilon[/tex] (fix a point [tex]t \in I[/tex] and an [tex]\epsilon>0[/tex]; then the [tex]\delta[/tex] corresponding to [tex]x \in I[/tex] is defined to be the value of the gauge at [tex]x[/tex]). From this particular gauge many standard theorems can be proved, e.g., the intermediate value theorem or the fact that a continuous function on a closed, bounded interval is uniformly continuous. Most remarkable is how short the proofs are. Many are nearly immediate, at least compared to the standard proofs I've been exposed to before.
I've found this approach really interesting and have been looking for more resources on it. I wonder if continuity can be defined in terms of gauges. And I wonder how well this generalizes (say to metric or topological spaces).
Anyone know about this? Thanks!
Edit:
I'm wanting to find as many resources on this since it seems to be a relatively unexplored topic and am planning to author an expository article discussing it, unless to my surprise I find many such articles which heretofore appear to be nonexistent.
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