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For example, take the function f(x) = k, for x in Z

Then based on the epsilon delta definition of a limit, for any epsilon > 0, we can always find a delta, for which 0 < |x-x_0| < delta implies |f(x)-k| = 0 < epsilon. Thus, the limit of every non-accumulation point of f(x) has limit = k.

This example seems to contradict the fact that limits are undefined at non-accumulation points.