0.1 Proposition Let P1 , Q1 , P2 , Q2 ∈ R[x] be polynomials such that Q1 6= 0
and Q2 6= 0 and let R̂ := R ∪ {∞} where ∞ denotes the unsigned point
at infinity. Then the function f1 : {x ∈ R : Q1 (x) 6= 0} → R given by
f1 (x) := P1 (x)/Q1 (x) has a unique continuous extension fˆ1 : R̂ → R̂.
Moreover the following are equivalent where f2 is defined similarly:
(i) There exists a countable subset S ⊂ dom(f1 ) ∪ dom(f2 ) such that for
every x ∈ S we have f1 (x) = f2 (x).
(ii) For every x ∈ dom(f1 ) ∪ dom(f2 ) we have f1 (x) = f2 (x).
(iii) P1 /Q1 = P2 /Q2 ∈ R(x), i.e., P1 Q2 = P2 Q1 .
0.2 Definition A subset I ⊂ R is said to be connected or an interval if and
only if for all x, y, z ∈ R such that x < y < z and x, z ∈ I, we have y ∈ I.
0.3 Proposition Let I, J ⊂ R be two intervals and let f : I → J be a function.
If any two of the following statements are true, then so is the third.
(i) f is continuous.
(ii) f is strictly monotone.
(iii) f is a bijection.
In this case the function f −1 : J → I inverse to f is also continuous and
strictly monotone.
0.4 Proposition Let a, b ∈ R satisfy a < b and let f : [a, b] → R be a function.
Suppose that f is differentiable on the interior (a, b) and that the limit
A :=
lim
f 0 (x) exists in R. Then f is differentiable at a if and only
x→a;x∈(a,b)
if f is continuous at a. In this case we have f 0 (a) = A.
0.5 DefinitionFor each subset X ⊂ R, we define the derived set X 0 of X
by X 0 := x0 ∈ R : inf 0 |x0 − x| = 0 . Members of X 0 also are called
x∈X\{x }
limit points of X in R.
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