Download Limits and Continuous Functions: Handout for Math 554 - Fall 2008 - Prof. R. Sharpley and more Study notes Mathematics in PDF only on Docsity!
Math 554
Limits and Continuous Functions
Handout #5 – Fall 2008
We need to recall the following:
Earlier Definitions
- A point x 0 is called a limit point of a set A if each nbhd of x 0 contains a member of A different from x 0 , i.e. for each ǫ > 0, (Nǫ(x 0 ){x 0 }) ∩ A 6 = ∅.
- A point x 0 ∈ A is called an isolated point of A if x 0 belongs to A but is not a limit point.
Earlier Theorems
- A set F is closed if and only if it contains all its limit points.
- x 0 is a limit point of a set A if and only if there exists a sequence {xn} ⊆ A such that xn → x 0 , but xn 6 = x 0 , (∀n ∈ IN ).
With these ideas we begin the study continuity of functions, which is very nat- urally framed in a general metric space.
Defn. Suppose that x 0 is a limit point of the domain of a function f : A → B, then f is said to have a limit L as x approaches x 0 if,
∀ǫ > 0 , ∃δ > 0 ∋ (x ∈ dom(f ) & 0 < dA(x, x 0 ) < δ) =⇒ dB(f (x), L) < ǫ.
In this case, we use the notation,
lim x→x 0 f (x) = L.
Defn. Suppose f: A → B, metric spaces. If x 0 ∈ A, then f is said to be continuous at x 0 if for each ǫ > 0 there is a δ > 0 so that if x ∈ A and dA(x, x 0 ) < δ, then dB(f (x), f (x 0 )) < ǫ.
Note. Remember that a point x 0 ∈ A is either an isolated point of A or it is a limit point of A. Considering these two cases separately, the definition for continuity of f at a point can be seen to be equivalent to the following in each of the respective situations:
- if x 0 is an isolated point of A, then f is automatically continuous. or
- if x 0 is a limit point of the domain A, then the condition lim x→x 0 f (x) = f (x 0 ) must hold.
Defn. Consider a set B∗^ ⊆ B. A set O˜ ⊆ B∗^ is called open relative to B∗^ (or briefly, relatively open) if O˜ = O ∩ B∗^ for some open set O ⊆ B. That is to say, O˜ is just the restriction to B∗^ of an open set in the whole space.
Theorem. Let f: A → B, where A, B are metric spaces, then TFAE
a.) f is continuous at each point of its domain, b.) for each limit point x 0 of the domain A, lim x→x 0
f (x) exists & equals f (x 0 ). c.) for each sequence xn → x 0 , then f (xn) → f (x 0 ) must hold, d.) f −^1 [O] is open for each open subset O of B.
We will be concentrating on the metric space of real numbers.
Note. The same proof above for continuity at a point x 0 can be used to show the corresponding result for limits holds. The only difference is that f is not required to be defined at x 0. The statement reads as:
Suppose that f: A → B is a real-valued function of a real variable, i.e. A, B ⊆ R. If x 0 is a limit point of the domain of f , then TFAE :
a.) lim x→x 0 f (x) = L, b.) For every sequence {xn} in the domain of f , if xn → x 0 , then f (xn) → L.
Corollary. The finite sum, product, or the quotient of continuous functions is each continuous on their respective domains.
Corollary. All polynomials are continuous. Rational functions are continuous on their domains.
Theorem. The composition of continuous functions is continuous.
Example. Each of the following are examples of continuous functions on their respective domains:
- f (x) := |x|
- g(x) :=
x
- F (x) :=
x^2 − 2 x + 5 x^3 − 1
