Functions and Sets

Question 1

What is the definition of a continuous function?

Answer 1

A function is continuous at x₀ iff lim_{x –> x0} f(x) = f(x₀) where x₀ is an interior point of the function’s domain
A function is continuous everywhere if it is continuous at any point in its domain
A function can be left continuous and right continuous (limit holds when approaching from one side)
If a function defined on [a, b] is continuous for all x₀ in (a, b), is left-continuous at b and right-continuous at a, then f(x) is continuous on [a, b]

Question 2

What is the definition of an interior point?

Answer 2

There exists some small open interval (a, b) that is a subset of domain D such that x is an element of (a, b)

Question 3

What is the definition of a differentiable function?

Answer 3

A function is differentiable at x₀ iff lim_{x –> x0} (f(x) - f(x₀))/(x - x₀) exists

Question 4

What is the relationship between differentiability and continuity?

Answer 4

Differentiability is sufficient but not necessary for continuity

Question 5

What are the rules for limits?

Answer 5

The limit of the sum and product of functions is the sum and product of the limits of the functions, the limit of a function raised to the power is the limit raised to that power

Question 6

When is a composite function differentiable?

Answer 6

(g o f)(x) is differentiable at a if f is differentiable at a and g is differentiable at f(a)

Question 7

When does the inverse of a function exist?

Answer 7

When the function is one-to-one

Question 8

What is the definition of a convex function?

Answer 8

f defined over [a, b] is convex over [a, b] if for all x, y in [a, b] and for all λ in (0, 1) f(λx + (1 - λ)y) ≤ λf(x) + (1 - λ)f(y)
i.e. f at the weighted average of x and y is (weakly) below the weighted average of f(x) and f(y)
f convex over (a, b) iff f’‘(x) ≥ 0 for all x in (a, b)

Question 9

What is the norm of x ∈ R²?

Answer 9

||x|| = sqrt(x₁² + x₂²)

Question 10

What is the n dimensional Euclidean space?

Answer 10

The set Rⁿ equipped with notions of norm and operations of summation and multiplication by scalars

Question 11

What is the definition of the epsilon neighbourhood of a point c in n dimensional Euclidean space or the open ball with centre c and radius ε?

Answer 11

B(c, ε) = {x in Rⁿ:||x - c|| < ε}

Question 12

What is the definition of an interior point of the subset S of Rⁿ?

Answer 12

s is an interior point of S if there exists an open ball with centre s and positive radius ε contained in the set S

Question 13

What is the complement of a subset S of Rⁿ?

Answer 13

S^C = Rⁿ\S = {x in Rⁿ|x not in S

Question 14

What is the definition of a boundary point?

Answer 14

b is a boundary point of set S if at every neighbourhood of b there are elements of S and S^C

Question 15

What is the definition of a closed set?

Answer 15

A set is closed iff it contains all of its boundary points
S is closed iff S^C is open

Question 16

What is the definition of a bounded set?

Answer 16

S is bounded iff there exists B such that for any x in S, ||x|| ≤ B

Question 17

What is the definition of a compact set?

Answer 17

A compact set is closed and bounded

Question 18

What is the definition of a convex and strictly convex set?

Answer 18

S is convex iff for any x, y in S and for any λ in [0, 1], λx + (1-λ)y is in S
Strictly: λx + (1-λ)y is in the interior of S

Question 19

What is the definition of a continuous multivariate function?

Answer 19

For a compact domain, f is continuous on D if it is continuous at each interior point of D and the limit as x approaches b of f(x) is f(b) for any boundary point b in D where the limit is taken so that x remains in D