Why does this function make it easy to prove continuity with sequences?

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Homework Help Overview

The discussion revolves around the continuity of a piecewise function defined differently for rational and irrational inputs. Participants are exploring the implications of using sequences to prove continuity at a specific point, particularly as the input approaches zero.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definition of convergence and how it relates to proving that the function approaches a limit as the input sequence tends to zero. Questions arise about the clarity of the proof and the necessity of using precise definitions.

Discussion Status

The conversation is ongoing, with participants providing insights into the definitions of limits and continuity. Some suggest that a rigorous proof is necessary, while others express skepticism about the exercise's validity. There is no explicit consensus on the approach to take.

Contextual Notes

Participants note that the function is continuous only at zero and question the nature of the problem, suggesting it may be misleading or overly complex for a typical homework exercise.

  • #31
I thought the goal was showing continuity using ##[x_n \rightarrow x ]\rightarrow [f(x_n) \rightarrow x] ##
 
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  • #32
FactChecker said:
It's as simple and basic as that.
Furthermore, it's the real thing:wink:
PeroK said:
A simpler approach was simply to note that:
$$|f(x) - f(0)| = |f(x)| \le |x|$$
Definitively, compañero.
WWGD said:
I thought the goal was showing continuity using ##[x_n \rightarrow x ]\rightarrow [f(x_n) \rightarrow x] ##
Yes.
Mark44 said:
This...
This... Was as simple as @PeroK knew; as hard as I turned it.
A thousand thanks, milesker, bihotzez.
HNY, PF!
 
  • #33
mcastillo356 said:
A thousand thanks, milesker, bihotzez.
Are the last two words Basque? I definitely don't recognize them as Spanish.
 
  • #34
J
Mark44 said:
Are the last two words Basque? I definitely don't recognize them as Spanish.
Just basqueing in the glory of Spanish regional languages.
 
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  • #35
In Basque, yes, "a thousand thanks, from the heart".
PD @FactChecker, I feel I should have written a few more about #30, but I've plunged headlong into the next chapter, integration.
Love, greetings, PF.
 
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