What Does It Mean for a Function to Be Continuous on a Curve?

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A function f is considered continuous on a curve γ if it can be expressed as the restriction of a continuous function F defined in a neighborhood of the curve. This definition implies that the pullback γ* f must also be continuous on the domain of γ. The discussion emphasizes the relationship between the continuity of functions and their behavior on curves, clarifying the mathematical framework involved.

PREREQUISITES
  • Understanding of continuous functions in topology
  • Familiarity with the concept of pullbacks in differential geometry
  • Basic knowledge of curves in mathematical analysis
  • Experience with neighborhoods in metric spaces
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  • Study the properties of continuous functions in topology
  • Learn about pullbacks and their applications in differential geometry
  • Explore the definition and examples of curves in mathematical analysis
  • Investigate neighborhoods and their significance in continuity
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T-O7
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Hey, I have a quick question: Does anyone know what it means for a function [tex]f[/tex] to be continuous on a curve [tex]\gamma[/tex]? Is it the same as saying the pullback [tex]\gamma^*f[/tex] is continuous on the domain of [tex]\gamma[/tex]?
 
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T-O7 said:
Hey, I have a quick question: Does anyone know what it means for a function [tex]f[/tex] to be continuous on a curve [tex]\gamma[/tex]? Is it the same as saying the pullback [tex]\gamma^*f[/tex] is continuous on the domain of [tex]\gamma[/tex]?



It implies that, actually. Off the top of my head, I'd say "continuous on a curve" means
that the function f is the restriction of a function F, continuous on a neighbourhood of the curve.
 

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