This function is more complicated. Consider the sequence = 1 / (2 n ). As n goes to infinity, the sequence converges to zero from the right. But f( ) = sin (2 n ) = 0 for all k. On the other hand, consider the sequence = 2 / ( (2n+1) ). Again, the sequence converges to zero from the right as n goes to infinity. But this time f( ) = sin( (2n+1) / 2) which alternates between +1 and -1. Hence, this limit does not exist. Therefore, the limit of f(x) as x approaches zero from the right does not exist.

Since f(x) is an odd function, the same argument shows that the limit of f(x) as x approaches zero from the left does not exist.

Therefore, the function has an essential discontinuity at x = 0.