Analysis ? Defn of the limit of a function?

Question

Give an example of a function f : (0,1)  R that has a limit at every point of (0,1) except ½. Use the definition of limit of a function to justify the example.

Answer 1

How about something simpler, like

f(x) = 0 when x<1/2, 1 when x>=1/2

Then the limit from the left is 0, the limit from the right is 1, so the regular limit doesn't exist.

Answer 2

tan(x)/pi

The limit of a function as you approach from both f-df and f+df is the same for all points within [0,1] except at 1/2 where the the limit of f-df approaches -inf and the limit of f+df approaches +inf.