The graph is not needed to see that #f# is continuous at #3#. The definition of "continuous at #3#" is #lim_(xrarr3)f(x) = f(3)#. And we are told that this is true.
The graph of #f# must include the point #(3,1)# and for values of #x# near #3#, the corresponding #y# values must be close to #1`#.
Another way to say this is: near #x=3#, the graph must be connected (on both sides) to #(3,1)#.
Reading from left to right:
The first graph does not have #f(3) = 1# (There is an open hole at #(3,1)#)
It also does not have #lim_(xrarr3) f(3) = 1# (The limit is #1# from the left and #-1# from the right.
The second graph has #f(3) = 1#, but again, it has limit #1# only from the left.
The third graph does have #lim_(xrarr3)f(x) = 1#, but the hole at #(3,1)# indicates that #f(3) != 1#. (In fact #f(3) does not exist on the third graph.)
Only the fourth graph has the two required properties.
There are infinitely many examples of functions that have 