COMPACT SETS, CONNECTED SETS AND CONTINUOUS FUNCTIONS
COMPACT SETS, CONNECTED SETS AND CONTINUOUS FUNCTIONS
Ali Scagnelli
Fall 2017
Properties of Continuous Functions
The definition below states the most basic yet very important properties of continuous functions. If we have to continuous functions and there is a constant c that is in R, and both f and g are continuous at x0 then the multiplication of c times f or g, addition of f + g, and fg are all continuous at x0 as well.
Let f,g : A → R and let c be an element of R. Suppose f and g are continuous at x0 which is an element of A. Then cf, f + g and fg are continuous at x0. Furthermore, if g(x0) does not equal 0, then f/g is continuous at x0.
Another important definition to know and understand is that of a uniformly continuous function. A function f is uniformly continuous on I when we are able to select δ independently of x0.
Let f be defined on a set A ⊂ R. We say that f is uniformly continuous (on A) if for every ε > 0 there exists δ > 0 such that if x,y ∈ A and |x−y| < δ, then |f(x) − f(y)| < ε.
When the interval I is (closed interval) [a,b] then every function f that is continuous on I is uniformly continuous on I.
Let f be continuous on [a, b]. Then f is uniformly continuous.
We can then use this theorem to prove that any continuous function on a closed bounded interval [a,b] is bounded.
Let f be continuous on [a, b]. For us to be able to find and absolute maximum and absolute minimum, f must be continuous on a closed and bounded set. This is supported by the example below where we try to substitute a closed interval with an open interval, and a bounded closed interval with an unbounded closed interval
f(x)=1/x ∈ (0,1)
f(x)= x for x ∈ [0,∞)
https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch3.pdf
This online resource was very helpful with understanding the topic of continuous functions on a compact domain. It comes from the University of California, Davis intro to analysis course. Much of its notation is similar to our textbook. It provided definitions and in depth examples that help better understand this topic of distinguishing whether or not functions are continuous.
Problem on Uniform Continuity see exercise 15.9 http://faculty.atu.edu/mfinan/3203/sol15.pdf
Darboux Property states that if the graph has no point on some horizontal line y = c, then the graph must be entirely above or below that line. In logical terms it is expressed as Theorem 5.53: Let f be continuous on [a,b] and let c ∈ R. If for every x ∈ [a,b], f(x) ̸= c, then either f(x)>c for all x∈[a,b] or f(x)<c for all x∈[a,b].
1. Definitions
1.1. D ⊂ R is compact if and only if for any given open covering of D we can subtract a finite sucovering. That is, given (Gα)α ∈ A a collection of open subsets of R (A an arbitrary set of indices) such that D ⊂∪α∈AGα, then there exists finitely many indices α1, . . . , αN ∈ A such that D ⊂ ∪N i=1Gα1 .
1.2. Let D an arbitrary subset of R. Then A ⊂ D is open in D (or relative to D, or D-open) if and only if there exists G open subse