Suppose f(x)=x+2, then f(1)=3 and point (1,3) is on the graph. Also as x→1, f(x)→3.
If f(x)=2x²-x+2, then f(1)=3, point (1,3) is on the graph and as x→1, f(x)→3.
In each case the graph is continuous.
But if f(x)=
{ 3 if x=1
{ x-1 if x≠1
f(1)=3 but if x→1, f(x)→0. Therefore (1,3) is on the graph but when x→1, f(x) does not approach 3, because the graph is discontinuous.
If f(x)=(x²+x-2)/(x-1), as x→1, f(x)→3, but f(x) is not defined for x=1, so (1,3) is not on the graph.