Can any one example a function that has limit just in one point

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A function defined as f(x)=x for rational x and f(x)=0 for irrational x has a limit only at x=0, where the limit is 0. To demonstrate that it lacks a limit at any point c≠0, one can use sequences approaching c: one sequence of rational numbers and another of irrational numbers. The sequential criterion for limits shows that these sequences yield different limit values, confirming the absence of a limit at c≠0. An epsilon-delta argument can also be applied, particularly noting that the behavior of irrational numbers affects the limit. This function serves as a clear example of a limit existing solely at one specific point.
hadi amiri 4
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can anyone example a function that has limit just in one point
 
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Let f(x)=x if x is rational and f(x)=0 if x is irrational.
 


can you prove it
 


Yup. Pretty easy to see that \lim_{ x \to 0 } f( x ) = 0, so we just want to show that it doesn't have a limit anywhere else. To show that it doesn't have a limit at c\neq0, take a two sequences approaching c, one along rational numbers and one along irrational numbers, and use the sequential criterion for limits. If you have trouble with that, post what you've tried and I'll fill in the details.
 


You could also use epsilon delta. Only numbers for which e > 0 might not be true are irrational x, but then you can just choose d = e.
 

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