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.
