Define:

where denotes the greatest integer function, or floor function. Sketch the graph of for and . Evaluate

Is it possible to define in a way that makes continuous at 0.

First, we sketch the graph of on the requested intervals.

As , alternates between and .

As , alternates between and .

There is no way to define to make continuous at 0 since will take both values and no matter how small we choose our . (So, if we were to try to define , then for there is no such that whenever , and similarly if we try to define .)