## 6.3. Discontinuous Functions

### Examples 6.3.4(c):

This function is more complicated. Consider the sequence*x*. As

_{n}= 1 / (2 n )*n*goes to infinity, the sequence converges to zero from the right. But

*f( x*for all

_{n}) = sin(2 n ) = 0*k*. On the other hand, consider the sequence

*x*. Again, the sequence converges to zero from the right as

_{n}= 2 / ( (2n+1) )*n*goes to infinity. But this time

*f( x*which alternates between

_{n}) = sin( (2n+1) / 2)*+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*.