1 - Limits and Continuity of Functions

Question 1

Define punctured neighbourhood

Answer 1

D is a punctured neighbourhood of c ∈ R if
there exists δ > 0 such that (c − δ, c) ∪ (c, c + δ) ⊂ D.

Question 2

ε-δ definition of a limit

Answer 2

Let f:D→R, c ∈ R, and D be a punctured neighbourhood of c. Then
[lim{x→c} f(x)] = L if for any ε > 0 there exists δ > 0 such that ∀x ∈ D, 0 < |x − c| < δ ⇒ |f(x) − L| < ε

Question 3

Define Inertia

Answer 3

Let f:D→R, D ⊂ R, and c, L, M ∈ R. Suppose D contains a punctured neighbourhood of c and that [lim{x→c}f(x)] =
L > M.
Then there exists δ > 0 such that ∀x ∈ D, 0 < |x − c| < δ ⇒ f(x) > M.

Question 4

Sequential characterisation of the limit

Answer 4

Let f : D → R, c, L ∈ R, and D be a punctured neighbourhood of c. Then
[lim{x→c}f(x)] = L is equivalent to:
For any sequence (x_n) ⊆ D \ {c} such that [lim{n→∞} x_n] = c, [lim{n→∞} f(x_n)] = L

Question 5

Define one-sided limit from the right (x→c+)

Answer 5

If there exists δ_0 > 0 such that (c, c + δ_0) ⊂ D, then for any ε > 0, there exists δ > 0 such that ∀x ∈ D, 0 < x−c < δ ⇒ |f(x) − L| < ε.

Question 6

Define one-sided limit from the left (x→c-)

Answer 6

If there exists δ_0 > 0 such that (c − δ_0, c) ⊂ D, then for any ε > 0, there exists δ > 0 such that ∀x ∈ D, −δ < x−c < 0 ⇒ |f(x) − L| < ε

Question 7

Define what it means for f:D→R to be continuous at c ∈ D.

Answer 7

For any ε > 0 there exists δ > 0 such that ∀x ∈ D, |x−c| < δ ⇒ |f(x) − f(c)| < ε.

Question 8

Define what it means for f:D→R to be continuous on D.

Answer 8

f is continuous at each point in D, i.e, for any ε > 0 there exists δ > 0 such that ∀x,y ∈ D, |x−y| < δ ⇒ |f(x) − f(y)| < ε.

Question 9

Sequential characterisation of continuity

Answer 9

Let f: D→R, c ∈ R, and D ⊂ R. Then f is continuous at c if and only if for all sequences (x_n) ⊆ D \ {c} such that [lim{n→∞} x_n] = c, then [lim{n→∞} f(x_n)] = f(c)

Question 10

Define the composition of functions f◦g.

Answer 10

If g: D→R and f: E→R are such that g(x) ∈ E for all x ∈ D, then, f◦g : D→R is defined by
(f◦g)(x) = f(g(x)), ∀x ∈ D

Question 11

What are the conditions for the composition of functions f◦g to be continuous at c?

Answer 11

1. g: D→R and f: E→R are such that g(x) ∈ E for all x ∈ D. 2. For c ∈ D, g is continuous at c and f is continuous at g(c).

Question 12

Intermediate Value Theorem

Answer 12

Let:
1. a, b ∈ R satisfy a < b,
2. f: [a, b]→R be continuous on [a, b]
3. f(a) < f(b).
Then, for all y ∈ (f(a), f(b)) there exists c ∈ (a, b) such that f(c) = y.

Question 13

What does it mean for a function f: D→R to be bounded above?

Answer 13

There exists an M ∈ R such that f(x) ≤ M, ∀x ∈ D.

Question 14

What does it mean for a function f: D→R to be bounded below?

Answer 14

There exists an m ∈ R such that f(x) ≥ m, ∀x ∈ D.

Question 15

What does it mean for a function f: D→R to be bounded?

Answer 15

There is an M > 0 such that
|f(x)| ≤ M, ∀x ∈ D

Question 16

Weierstrass Extremal Value Theorem

Answer 16

Let f: [a, b]→R be continuous on [a, b]. Then f is bounded and attains its supremum and infimum.
i.e, there exists p, q ∈ [a, b] such that:
sup f [a,b] = f(p), and
inf f [a,b] = f(q)

Question 17

Define inverse of a function f^{-1}.

Answer 17

Let f: I→J be a bijection. Then f^{−1}: J → I is defined as
f^{−1}(y) = x where y ∈ J and
x ∈ I is the unique element in I such that f(x) = y.