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
.)