Compare and contrast continuity of a function?

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SUMMARY

The discussion focuses on analyzing the continuity of the piecewise function defined as f(x) = { sin(1/x) when x ≠ 0, 0 when x = 0. The key conclusion is that the function is not continuous at x = 0 because the limit of sin(1/x) as x approaches 0 does not exist, leading to a discontinuity at that point. The value of the function at x = 0 is defined as 0, but the behavior of sin(1/x) creates a gap, confirming the lack of continuity.

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javier007
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PLEASE help me. I need to analyze the continuity of the piecewise function

f(x) = { sin(1/x) when x is not = to 0
_____{ 0_____ when x = 0

so i know sin(1/x) doent have a value at 0 but the second part of the function places the value of 0 at 0...BUT are both parts connected without any gaps, holes, or jumps in bewteen? in other words is it continuous or not?

I would appreciate it A LOT if you could explain it to me why or why not

thanks
 
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