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

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The discussion centers on understanding the continuity of the function defined as f(x) = |x| if x is rational and 0 if x is irrational, particularly as x approaches 0. Participants emphasize that to prove f(x_n) converges to 0 for any sequence x_n tending to 0, one must apply the formal definition of limits and continuity. They highlight that both rational and irrational sequences approaching 0 yield the same limit of 0 for f(x_n). The conversation also touches on the need for a rigorous proof rather than informal reasoning, suggesting that a comprehensive approach should consider arbitrary sequences. Ultimately, the function's continuity is confirmed only at the point 0, illustrating the complexity of piecewise functions in limit proofs.
  • #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.
 
  • Like
Likes mcastillo356
  • #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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