Continuity Flashcards
Continuity at a point
Let f: X -> R be a real function and suppose that a e X. Then f i continuous at a, if given any e in R+, there exists d in R+ such that
|f(x)-f(a)| < e whenever x e X and |x-a|
Continuity in a neighbourhood of a
If f(x) is defined for all x in a neighbourhood of a ((a-R, a) U (a, a+R)), then f is continuous at a iff lim, as x tends to a, f(x) = f(a)
Continuity at end points
If f is defined on an interval [a,b] where a,b e R, then
(1) f is continuous at the end point a iff lim, x tends a+, of f(x) = f(a)
(2) f is continuous at b iff lim, x tends to b-, of f(x) = f(b).
Algebra of continuous functions at a
Suppose that f: X-> R and g: X-> R are continuous at a point a e X. Then
(1) f+g is continuous at a
(2) f-g is continuous at a
(3) fg is continuous at a
(4) f/g is continuous at a
f is continuous
Let f: X-> R be a real function. Then f is continuous if f is continuous at every point in X.
Algebra of continuous functions
Suppose that f: X-> R and g: X-> R are continuous. Then
(1) f+g is continuous
(2) f-g is continuous
(3) fg is continuous
(4) f/g is continuous
Composition
Suppose that f: X -> Y and g: Y-> Z are functions. Then the composition of g.f : X-> Z is defined by
g.f(x) = g(f(x)) for all x e X.
Functions composed of continuous functions are continuous (at a)
Suppose that f: X-> Y and g: y-> Z are functions and that f is continuous at a point a e X, and that g is continuous at the point f(a) e Y. Then the composed function g.f : x-> Z is continuous at a.
Functions composed of continuous functions are continuous
Suppose that X and Y are intervals and that f : X-> Y and g: Y->|R are continuous functions. Then the composed function g. f : X -> |R is continuous
Continuous functions cannot instantaneously jump
Suppose that f : [a,b] -> |R is continuous and # e R. If f(w) > # for some w e [a,b], then there exists d in R+ such that f(x)># whenever x e [a,b] and |x-w|< d.
Continuous functions cannot instantaneously jump at end points
Suppose that f: [a,b] -> |R is continuous and # e R. If f(x) >= # for all x e (a,b) then f(a)>= # and f(b) >= #
Intermediate value theorem
Suppose that f: [a,b] -> |R is continuous, and that f(a)=/ f(b), and that # lies strictly between f(a) and f(b). Then there exists c in (a,b) s.t. f(c) = #.
Finding the roots of equations
Use the IVT to prove there is a root of the equation
(1) Prove that the function is continuous
(2) Find a interval [a,b] s.t. f(a)<0
Fixed point theorem
If f:[a,b] ->[a,b] is continuous, then there exists c e [a,b] s.t. f(c) =c.
Least upper bound axiom
Any bounded set X of real numbers has a least upper bound
Monotone convergence theorem
Any monotone bounded sequence (xn) of real numbers is convergent
Bolzano-Weierstrass
Any bounded sequence (xn) of real numbers has a convergent subsequence.
Bounded
A function f: X-> R is bounded is bounded if there is a number R in |R+ such that | f(x) | =< R for all x e X.
A bounded function attains its bounds
A bounded function f: X-> R attains its bounds if there exists points c and d in X such that
f(c) = sup f(x) and f(d) = inf f(x)
Boundedness theorem
A continuous function on a closed bounded interval is bounded and attains its bounds.
Image of a closed bounded interval
Suppose that f:[a,b] -> |R is continuous. Then f([a,b]) is a closed bounded interval. More precisely
f( [a,b] ) = [m, M]
where m= sup f( [a,b] ) and m = inf f( [a,b]).
Injective
A real function f: X -> Y is injective if f(x1) =/ f(x2) whenever x1, x2 e X and x1=/ x2.
Equivalently, there exists at most one x e X such that f(x) =y.
Surjective
The function f is surjective if, for each y e Y, there exists at least one x e X such that f(x) =y.
Bijective
The function f is bijective if it is both injective and surjective. That is, if for each y e Y, there is exactly one x e X such that f(x) = y.
Inverse function
Let f : X-> Y be a bijection. The inverse function f^-1: Y-> X is defined by
f^-1(y) = x
for each y e Y, where x is the unique element of X such that f(x) = y.
Strictly increasing
A real function f: X -> Y is strictly increasing if
f(x1) < f(x2)
whenever x1, x2 e X and x1< x2.
Strictly decreasing
A real function f: X-> Y is strictly decreasing if
f(x1)>f(x2)
whenever x1, x2 e X and x1< x2.
Strictly monotone
A real function is strictly monotone if it is either strictly increasing or strictly decreasing.
Inverse function theorem
Suppose that f: [a,b] -> [c,d] is continuous and strictly increasing, that f(a)=c and f(b) = d. Then the inverse function f^-1 : [c,d] -> [a,b] exists, is continuous, strictly increasing and surjective.
