For example, the integer part function is right continuous but not left continuous at 1 (v.s. figure fig integer part 2).

- the function is left-continuous at ;
- the function is right-continuous at .

- (i)
- If , then . It follows that ; thus is left continuous at 0.
- (ii)
- If , then . It follows that ; thus is right continuous at 0.

Let us compute the one-sided limits of at 0:

As , this shows that is left-continuous at 0 and right-continuous at 0, thus is continuous at 0.

and |

As we know that , we conclude that is right-continuous at but not left-continuous at .

Noah Dana-Picard 2007-12-28