  The subject matter of this work is the set of integral points(i.e. points
with both coordinates integers) on the graphs of rational functions of the form
f(x)=(x^2+bx+c)/(x+a), with a,b,c,being integers.Following the introduction, we
establish Proposition1 in Section2. This proposition plays a key role in the
proof of Theorem1 in Section5. Proposition1 is proved with the aid of Euclid's
lemma and another well known result in number theory; see reference [1].In
Sections3 and4, we focus on the special case b^2-4c=0; which implies b=2d and
c=d^2, for some integer d. If d and a are distinct; then there are finitely
many integral points, parametrically described in Results1 and2. Theorem1 in
Sec.5 deals with the general case.Accordingly, if a^2-ab+c is not zero; there
are exactly 4N distinct integral points parametrically described.Except in the
cases where a^2-ab+c is a perfect square, or minus a perfect square; in which
cases thare are exactly 4N-2 distinct integral points. Here N stands for the
number of positive divisors, not exceeding the square root of the absolute
value of a^2-ab+c. (divisors of that absolute value).The paper concludes with
Th.2, which is a direct application of Th.1 in the cases where the above
absolute value is 1, p, or p^2; p a prime.
