6.3. Discontinuous Functions
Which of the following functions, without proof, has a 'fake' discontinuity,
a 'regular' discontinuity, or a 'difficult' discontinuity ?
This function seems to have a 'fake' discontinuity at x = 3
we could easily move the single point at x = 3
to the right height,
thereby filling in the discontinuity.
This function, while simple, seems to have a 'true' discontinuity
at x = 0
. We can not change the function at a single point to
make it continuous.
This function is unclear. It is hard to determine what exactly
is going on as x
gets closer to zero. Assuming that the function
does turn out to be discontinuous at x = 0
, it definitely seems
to have a 'difficult' discontinuity at x = 0.
This function is impossible to graph. The picture above is only
a poor representation of the true graph. Nonetheless, given any
, the function jumps between 1 and 0 in every neighborhood
. That seems to mean that the function has a difficult discontinuity
at every point.