Exam Revision

Question 1

A function f is continuous at c ∈ dom(f) if

Answer 1

∀ε > 0, ∃δ > 0 s.t. ∀x ∈ dom(f), |x − c| < δ =⇒ | f(x) − f(c)| < ε.

Question 2

A set A ⊆ R is bounded above iff

Answer 2

∃M ∈ R s.t. ∀a ∈ A, a ≤ M

Question 3

A set A ⊆ R is bounded below iff

Answer 3

∃m ∈ R s.t. ∀a ∈ A, m ≤ a.

Question 4

Bounded iff

Answer 4

bounded above and bounded below

Question 5

A set A ⊆ R is bounded iff

Answer 5

∃K > 0 s.t. ∀a ∈ A, |a| ≤ K.

Question 6

Least upper bound iff

Answer 6

∀ε > 0, ∃a ∈ A s.t. a > M − ε

Question 7

Subsequence

Answer 7

A subsequence of a sequence (xn)∞n=1
is a sequence of the form (xkn)∞n=1
for some strictly increasing natural numbers k1 < k2 < k3 < . . . .

Question 8

Theorem 5.90 (Algebraic properties of continuity). Let f : dom(f) → R and g : dom(g) → R be real functions and λ ∈ R. Moreover let c ∈ dom(f) ∩ dom(g) and assume that f and g are continuous at c. Then

Answer 8

a) f + g is continuous at c;
b) λ f is continuous at c;
c) f g is continuous at c;
d) If g(c) ̸= 0, then f /g is continuous at c