Continuity

Question 1

Continuity at a point

Answer 1

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|

Question 2

Continuity in a neighbourhood of a

Answer 2

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)

Question 3

Continuity at end points

Answer 3

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).

Question 4

Algebra of continuous functions at a

Answer 4

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

Question 5

f is continuous

Answer 5

Let f: X-> R be a real function. Then f is continuous if f is continuous at every point in X.

Question 6

Algebra of continuous functions

Answer 6

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

Question 7

Composition

Answer 7

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.

Question 8

Functions composed of continuous functions are continuous (at a)

Answer 8

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.

Question 9

Functions composed of continuous functions are continuous

Answer 9

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

Question 10

Continuous functions cannot instantaneously jump

Answer 10

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.

Question 11

Continuous functions cannot instantaneously jump at end points

Answer 11

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) >= #

Question 12

Intermediate value theorem

Answer 12

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) = #.

Question 13

Finding the roots of equations

Answer 13

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

Question 14

Fixed point theorem

Answer 14

If f:[a,b] ->[a,b] is continuous, then there exists c e [a,b] s.t. f(c) =c.

Question 15

Least upper bound axiom

Answer 15

Any bounded set X of real numbers has a least upper bound

Question 16

Monotone convergence theorem

Answer 16

Any monotone bounded sequence (xn) of real numbers is convergent

Question 17

Bolzano-Weierstrass

Answer 17

Any bounded sequence (xn) of real numbers has a convergent subsequence.

Question 18

Bounded

Answer 18

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.

Question 19

A bounded function attains its bounds

Answer 19

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)

Question 20

Boundedness theorem

Answer 20

A continuous function on a closed bounded interval is bounded and attains its bounds.

Question 21

Image of a closed bounded interval

Answer 21

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]).

Question 22

Injective

Answer 22

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.

Question 23

Surjective

Answer 23

The function f is surjective if, for each y e Y, there exists at least one x e X such that f(x) =y.

Question 24

Bijective

Answer 24

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.

Question 25

Inverse function

Answer 25

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.

Question 26

Strictly increasing

Answer 26

A real function f: X -> Y is strictly increasing if
f(x1) < f(x2)
whenever x1, x2 e X and x1< x2.

Question 27

Strictly decreasing

Answer 27

A real function f: X-> Y is strictly decreasing if
f(x1)>f(x2)
whenever x1, x2 e X and x1< x2.

Question 28

Strictly monotone

Answer 28

A real function is strictly monotone if it is either strictly increasing or strictly decreasing.

Question 29

Inverse function theorem

Answer 29

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.