(1) We sayis discontinuous at a point if it is not continuous at . This can happen in any of the following
ways:
(i) does not exist. The nonexistence of can happen in two ways :
(a) Essential discontinuity:
Either , or , or both does not exist. Such a
point of discontinuity is called essential discontinuity of. For example, for at , both the left and the right hand limit of at do
not exist. Thus, no value forwill makecontinuous at .
(b) Jump discontinuity:
If both and exist, but are not equal,
then such a point of discontinuity is called jump discontinuity of .
For example
has jump discontinuity at .
and 
.
is 
if it is not continuous at 
does not exist. The nonexistence of 
can happen in two ways :
, or 
, or both does not exist. Such a 
. For example, for
at 
, both the left and the right hand limit of 
do
will make
and 
exist, but are not equal, 
. 
. 
