6.3. Discontinuous Functions
Prove that f(x) has a discontinuity of second kind at x = 0This function is more complicated. Consider the sequence xn = 1 / (2 n ). As n goes to infinity, the sequence converges to zero from the right. But f( xn) = sin(2 n ) = 0 for all k. On the other hand, consider the sequence xn = 2 / ( (2n+1) ). Again, the sequence converges to zero from the right as n goes to infinity. But this time f( xn) = 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.