Calculus 1
Theorems and Definition: No Proof
Giovanni Michele Miranda
Contents
1 Relation of Equivalence and Sets
3
2 Supremum, Infimum, Maximum, Minimum
3
3 Supremum and Real Numbers: Properties
4
4 Limits
4
4.1
Limit of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . .
4
4.2
Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5 Definition of continuous function
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6 Weierstrass Theorem
5
7 Points of discontinuity of a function
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7.1
First kind: Jump Discontinuity . . . . . . . . . . . . . . . . . . .
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7.2
Second kind: Infinite Discontinuity . . . . . . . . . . . . . . . . .
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7.3
Third kind: Removable Discontinuity . . . . . . . . . . . . . . . .
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8 Derivatives and differentiable functions
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9 Derivatives: Non-Differentiable points
6
9.1
Corner Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
9.2
Vertical Tangent Point . . . . . . . . . . . . . . . . . . . . . . . .
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9.3
Cusp Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10 Fermat’s Theorem
6
1
11 Cauchy’s Theorem
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12 Convex functions and Second Derivatives
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13 Definition of Taylor Polynomials
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2
1
Relation of Equivalence and Sets
In Mathematics, a relation of equivalence is that ordered relation on U which
satisfies the three fundamental properties:
Reflexive: an ∼ an ;
Symmetric: if an ∼ bn then bn ∼ an ;
Transitive: if an ∼ bn and bn ∼ cn then an ∼ cn .
The ordered set (U, ≤) is said to be totally ordered if ∀x, y ∈ U either x ≤ y or
y ≤ x (i.e. any x ∈ U is comparable with any y ∈ U ). There are many ordered
universes that are not totally ordered, and they are called partially orderd set
(POS). Here are two examples:
• Take X ̸= ∅ and considerr P (X) := the set of all subsets of X. The order
relation we consider is ⊆. If X has at least two elements, (P (X), ⊆) is a
POS but not a totally ordered set. Indeed, if x ̸= y then they are two
elements of X where {x} ⊈ {y} and {y} ⊈ {x}.
• In N, say ’x < y’ if x divides y. It is easy to check that (N, <) is a POS.
Clearly it is not totally ordered: 5 < 7 and 7 ≮ 5 !
However, in this course we only met totally ordered universes, so from
this point on we assume, without saying, that all our ordered universes
are totally ordered.
2
Supremum, Infimum, Maximum, Minimum
Infimum (Inf ): It is the greatest lower bound, or the largest real number that
is less than or equal to every element in the set. It is denoted as inf(A), where
A is the set.
Supremum (Sup): It is the least upper bound, or the smallest real number
that is greater than or equal to every element in the set. It is denoted as sup(A),
where A is the set. 1
Maximum: it is the largest element in the set; for a set to have a maximum,
it must be a finite set. x ∈ A, is the maximum of A if ∀y ∈ A, y ≤ x.
Minimum: it is the smallest element in the set; for a set to have a minimum,
1 Supremum and infimum are defined for both finite and infinite sets; not every set has a
maximum or a minimum, but if a set has a supremum or infimum, it is unique. Supremum
and infimum may or may not be elements of the set, but this is not the case for maximum
and minimum.
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it must be a finite set. z ∈ A, is the minimum of A if ∀y ∈ A, y ≥ z.
Note: the difference between majorant/minorant and maximum/minimum is
that in the first cases we consider x/z ∈ U (so they can be out of A, which is the
definition set of maximum/minimum); so, in a certain way, majorant/minorant
are wider concepts.
3
Supremum and Real Numbers: Properties
Def: An ordered universe (U, ≤) has the Property of the supremum if any nonempty subset of U has the supremum. (R, +, · , ≤) is an ordered field with the
property of the supremum (i.e. any non-empty subset of R has the supremum
that can be a real number of +∞).
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Limits
4.1
Limit of a Sequence
Let {an }∞
n=1 be a sequence of real numbers.
• Finite Limit:
We say that l ∈ R is limit of {an } if ∀ϵ > 0 ∃nϵ s.t. ∀n ≥ nϵ , |l − an | < ϵ.
• Infinite Limit:
We say that +∞ or (−∞) is the limit of {an } if ∀M ∈ R ∃nM s.t. ∀ ≥ nM
then an > M .
An equivalent definition can be given in terms of neighbourhoods (not enunciated here).
4.2
Limit of a Function
Let f : Im (x0 ) − x0 → R where Im (x0 ) = (x0 − x, x0 + x).
• Finite Limit:
We say that limx→x0 f (x) = l ∈ R if ∀ϵ > 0, ∃sϵ > 0 s.t. ∀x ̸= x0 s.t.
|x − x0 | < sϵ so |f (x) − l| < ϵ.
• Infinite Limit:
We say that limx→x0 f (x) = +∞(or − ∞) if ∀M , ∃SM s.t. ∀x ̸= x0 with
|x − x0 | < SM , f (x) > M (or − ∞ respectively).
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5
Definition of continuous function
• Let f : Im (x0 ) → R; we say that f is continuous at x0 if limx→x0 f (x) =
f (x0 ).
• Let f : (a, b) → R; then f is continuous on (a, b) if it is continuous
at each point of (a, b).
• Let f : [a, b] → R; then f is continuous on [a, b] if it is continuous on (a, b)
and:
– limx→a+ f (x) = f (a).
– limx→b− f (x) = f (b).
So, f is continuous from the right at a, and from the left at b.
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Weierstrass Theorem
A real valued function f continuous on a closed and bounded interval [a, b] ⊆ R
has a maximum and a minimum on [a, b]. That is, there exist x1 , x2 ∈ [a, b]
s.t. ∀x ∈ [a, b] f (x1 ) ≤ f (x) ≤ f (x2 ).
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Points of discontinuity of a function
7.1
First kind: Jump Discontinuity
A function f : Ir (x0 ) → R continuous in Ir (x0 ) − x0 has a jump (or discontinuity of the first kind ) at x0 if limx→x± f (x) : ∃ and are finite, but
limx→x+ f (x) ̸= limx→x− f (x).
7.2
Second kind: Infinite Discontinuity
A function f : Ir (x0 ) → R continuous in Ir (x0 ) − x0 has a discontinuity of the
second kind at x0 if either limx→x+ f (x) or limx→x− f (x) (or both) is ∞ or ∄.
7.3
Third kind: Removable Discontinuity
A function f : Ir (x0 ) → R continuous in Ir (x0 ) − x0 has a removable point
of discontinuity at x0 if limx→x0 f (x) = l ∈ R but l ̸= f (x0 ).
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8
Derivatives and differentiable functions
Let f : Ir (x0 ) → R, we define the derivative of f at x0 the limit (if it ex(x0 )
(x0 )
ists): f ′ (x0 ) = limx→x0 f (x)−f
= limh→0 f (x0 +h)−f
. We say that f is
x−x0
h
differentiable at x0 if f ′ (x0 ) exists and is finite.
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Derivatives: Non-Differentiable points
9.1
Corner Point
Let f be continuous in Ir (x0 ) and differentiable in Ir (x0 ) − x0 . We say that
f has a corner at x0 if:
(x0 )
′
(x0 ) = limh→0+ f (x0 +h)−f
: ∃ and finite, and
• f+
h
(x0 )
′
(x0 ) = limh→0− f (x0 +h)−f
• f−
: ∃ and finite, but
h
′
′
(x0 ).
(x0 ) ̸= f−
f+
9.2
Vertical Tangent Point
Let f be continuous in Ir (x0 ) and differentiable in Ir (x0 ) − x0 . We say that
(x0 )
x0 is a point with vertical tangent for f if limx→x0 f (x)−f
= +∞(or − ∞);
x−x0
which means that: f ′ (x0 ) = +∞ or f ′ (x0 ) = −∞.
9.3
Cusp Point
Let f be continuous in Ir (x0 ) and differentiable in Ir (x0 ) − x0 . We say that
f has a cusp at x0 if:
′
′
(x0 ) = −∞, or
(x0 ) = +∞ and f−
• f+
′
′
• f+
(x0 ) = −∞ and f−
(x0 ) = +∞.
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Fermat’s Theorem
If f is continuous in Ir (x0 ), differentiable in x0 and f (x) ≤ f (x0 ) or f (x) ≥ f (x0 )
∀x ∈ Ir (x0 ) (i.e. if f has a local maximum or local minimum in x0 ) then
f ′ (x0 ) = 0, i.e. x0 is a stationary point for f .
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11
Cauchy’s Theorem
Let f and g be continuous real valued functions on [a, b], differentiable on (a, b)
′
(ξ)
(b)−f (a)
s.t. g ′ (x0 ) ̸= 0 ∀x ∈ (a, b). Then ∃ξ ∈ (a, b) s.t. fg′ (ξ)
= fg(b)−g(a)
.
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Convex functions and Second Derivatives
Def: f ∈ C 2 ((a, b)) is convex in (a, b) iff f ′′ (x) ≥ 0 ∀x ∈ (a, b).
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Definition of Taylor Polynomials
Let f ∈ C n (Ir (x0 )) (i.e. functions with n derivatives and continuous in Ir (x0 ))
and let 0 ≤ m ≤ n.
We define Pm f (x; x0 ) =
Pm
k=0
f n (x0 )
k
k! (x − x0 ) as the Taylor Polynomial of
order m of f, with center x0 .
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