7 - Functions

Question 1

Define limits for functions

Answer 1

We say that f(x) converges to b as x approaches x_0 if for every ε > 0 there exists 𝛿 > 0 such that

|f(x) - b| < ε

for all x in the domain with 0 < |x - x_0| < 𝛿. We write

f(x) –> b as x –> x_0 or b = lim(x->x_0) f(x)

Suppose that the domain is not bounded from above. Then b = lim(x->∞) f(x) if for every ε > 0 there exists 𝛿 > 0 such that

|f(x) - b| < ε

for all x in the domain with x > 𝛿

Question 2

What is continuity?

Answer 2

A function f:D —> {R} is said to be continuous at x_0 in the domain if lim(x->x_0) f(x) = f(x_0). Moreover, f is said to be continuous if it is continuous at every x in the domain.

Question 3

What does relatively open mean?

Answer 3

A set U is relatively open in D if U = ∅ or for every x in U there exists r > 0 such that B(x,r) \Intersection D is an improper subset of U

Question 4

What does relatively closed mean?

Answer 4

A set A is relatively closed in D if A^c \Intersection D is relatively open in D

Question 5

What is the intermediate value theorem?

Answer 5

The intermediate value theorem states that if f f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f ( a ) f(a) and f ( b ) f(b) at some point within the interval.

This has two important corollaries:

1. If a continuous function has values of opposite signs inside an interval, then it has a root in that interval (Bolzano’s theorem).
2. The image of a continuous function over an interval is itself an interval.

Question 6

What is the Bolzano-Weierstrass Theorem?

Answer 6

The theorem states that each infinite bounded sequence in {R}^n has a convergent subsequence. An equivalent formulation is that a subset of {R}^n is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem.

Question 7

What is a bounded set?

Answer 7

A set, A, is called bounded if there exists R >= 0 such that |x| <= R for all x in A.

Question 8

What is a sequentially compact set?

Answer 8

A set S is sequentially compact if every sequence in S has a subsequence that converges to something in S.

Question 9

What are closed and open sets?

Answer 9

A set is open if every point in is an interior point. A set is closed if it contains all of its boundary points.

Question 10

What is the continuity of inverse functions?

Answer 10

If f is injective (one-to-one) and continuous on an interval I, then the inverse function f^-1 exists and is continuous on a corresponding interval J (in the image or range of f).

Question 11

What is an image of a function?

Answer 11

The image of a function is the set of all output values it may produce.

More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the “image of A under (or through) f”.

Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B.

Question 12

What is the extreme value theorem?

Answer 12

The extreme value theorem (Weierstrass) states that if a real-valued function f is continuous on the closed interval [a, b], then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in [a, b] such that:

f(c) ≥ f(x) ≥ f(d), ∀x ∈ [a, b]

Question 13

What is uniform continuity?

Answer 13

Let D be a non-empty subset of R. A function f: D→R is called uniformly continuous on D if, for any ε > 0, there exists δ > 0 such that if u,v ∈ D and |u−v|< δ, then

|f(u)−f(v)| < ε

Question 14

How are uniform and ordinary continuity different?

Answer 14

The difference between uniform continuity and (ordinary) continuity is that in uniform continuity there is a globally applicable δ (the size of a function domain interval over which function value differences are less than ϵ) that depends on only ε, while in (ordinary) continuity there is a locally applicable δ that depends on both ε and x.

So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous.

Question 15

What is the mean value theorem?

Answer 15

The Mean Value Theorem (or Lagrange theorem) establishes a relationship between the slope of a tangent line to a curve and the secant line through points on a curve at the endpoints of an interval. The theorem is stated as follows.

If a function f(x) is continuous on a closed interval [a,b] and differentiable on an open interval (a,b), then at least one number c ∈ (a,b) exists such that

f’(c) = [ (f(b) - f(a)) / (b - a) ]

Question 16

What is a “sign” function?

Answer 16

The sign of a real number, also called sgn or signum, is -1 for a negative number (i.e., one with a minus sign “-“), 0 for the number zero, or +1 for a positive number (i.e., one with a plus sign “+”). In other words, for real x,

sgn(x) =
{-1 for x<0;
{0 for x=0;
{1 for x>0

Question 17

What are continuous functions?

Answer 17

A continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function.

This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument.

In dumbie terms, a function is continuous when its graph is a single unbroken curve that you could draw without lifting your pen from the paper.

Question 18

What are discontinuous functions?

Answer 18

A discontinuous function is a function that is not continuous. Just like a continuous function has a continuous curve, a discontinuous function has a discontinuous curve.

In other words, we can say that the graph of a discontinuous function cannot be made with a single stroke of the pen. A discontinuous function has breaks/gaps on its graph and hence, in its range on at least one point.

Question 19

What makes a function discontinuous?

Answer 19

Any functions whose graph contains holes, jumps, as asymptotes (vertical for domains, horizontal for ranges)

Question 20

What is a domain?

Answer 20

The domain of a function is the set of all possible inputs for the function.

Question 21

What are some cases of discontinuous functions?

Answer 21

A function f is said to be a discontinuous function at a point x = a in the following cases:

• The left-hand limit and right-hand limit of the function at x = a exist but are not equal.
• The left-hand limit and right-hand limit of the function at x = a exist and are equal but are not equal to f(a).
• f(a) is not defined.

Question 22

What are the types of discontinuous functions?

Answer 22

There are three types of discontinuities of a function - removable, jump and essential.

• Removable Discontinuity: For a function f, if the limit lim x→a f(x) exists (i.e., lim x→a- f(x) = lim x→a+ f(x)) but it is NOT equal to f(a). It is called ‘removable discontinuity’.
• Jump Discontinuity: For a function f, if the left-hand limit lim x→a- f(x) and right-hand limit lim x→a+ f(x) exist but they are NOT equal. Hence, the limit if the function f does not exist. Then, x = a is called ‘jump discontinuity’ (or) ‘non-removable discontinuity’.
• Essential Discontinuity: The values of one or both of the limits lim x→a- f(x) and lim x→a+ f(x) is ± ∞. It is called ‘infinite discontinuity’ or ‘essential discontinuity’. One of the two left-hand and right-hand limits can also not exist in such discontinuity.

Question 23

What are some properties of continuous functions?

Answer 23

i.) f, g - continuous at x_0, alpha is in {R}. Then:

• f+g, fg, alphaf, f-g are continuous at x_0.
• g(x_0) =/ 0 => f/g is continuous at x_0

ii.) The composition of continuous functions is continuous.

f - continuous at g(x_0)
g - continuous at x_0

=> Composition (f, g) is continuous at x_0

iii.) All elementary functions are continuous:

• constants
• power
• exponential
• logarithmic
• trigonometric
• inverse trigonometric and their compositions
• polynomials
• rational functions, etc.

Question 24

What is the connection between continuity and the limits of sequences?

Answer 24

The definition of continuity is given with the help of limits as, a function f with variable x is continuous at the point “a” on the real line, if the limit of f(x), when x approaches the point “a”, is equal to the value of f(x) at “a”, that means f(a).

lim(x->a) f(x) = f(a)

Question 25

What is Weierstrass theorem?

Answer 25

The extreme value theorem

Question 26

What is the Bolzano-Cauchy theorem?

Answer 26

The intermediate value theorem.

Question 27

What is the Bolzano theorem?

Answer 27

If a continuous function has values of opposite signs inside an interval, then it has a root in that interval