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

[hadi amiri 4]

can anyone example a function that has limit just in one point

 

[rochfor1]

Let f(x)=x if x is rational and f(x)=0 if x is irrational.

 

[hadi amiri 4]

can you prove it

 

[rochfor1]

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.

 

[snipez90]

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.