Victor Kreiman
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Math 350 | Advanced Calculus | Spring 2026

This study guide gives the main definitions and theorems covered in the course.

Contents

  1. The Real Numbers
  2. Cardinality
  3. Convergence of a Sequence
  4. Topology of \( \mathbb{R} \)
  5. Topology of Metric Spaces
  6. Limit of a Function
  7. Continuity
  8. Differentiation
  9. Integration
The Real Numbers

Consider the following sets of numbers: \begin{align} \mathbb{N} &= \{1, 2, 3, \ldots \}\ \text{ (natural numbers)}\\ \mathbb{Z} &= \{\ldots,-3,-2,-1,0,1, 2, 3, \ldots \}\ \text{ (integers)}\\ \mathbb{Q} &= \left\{\frac{a}{b} : a,b\in \mathbb{Z}, b\neq 0 \right\}\ \text{ (rational numbers)}\\ \end{align} We have that \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}\).

There does not exist \(p\in \mathbb{Q}\) such that \(p^2=2\).

In other words, \(\sqrt{2}\notin \mathbb{Q}\). We will shortly introduce the set of real numbers, \(\mathbb{R}\), a larger set of numbers than \(\mathbb{Q}\) which contains \(\sqrt{2}\). We will define \(\mathbb{R}\) rigorously as a set satisfying eleven axioms. The first ten of these axioms are given in the following three definitions.

A field is a set \(F\) together with two binary operations, denoted by \(+\) and \(\cdot\), which satisfy the following axioms:
  1. For all \(a,b\in F\), \(a+b = b+a\) and \(a\cdot b=b\cdot a\).
  2. For all \(a,b,c\in F\), \((a+b)+c = a+(b+c)\) and \((a\cdot b)\cdot c = a\cdot (b\cdot c)\).
  3. For all \(a,b,c\in F\), \(a\cdot (b + c) = a\cdot b + a\cdot c\).
  4. There exist elements \(0,1\in F\) for which \(0+a=0\) and \(1\cdot a = a\) for all \(a\in F\).
  5. For every \(a\in F\), there exists an element \(-a\in F\) for which \(a + (-a)=0\).
  6. For every \(a\in F\), \(a\neq 0\), there exists \(a^{-1}\in F\) for which \(a\cdot a^{-1}=1\).
Let \(S\) be any set. A total order (or linear order) on \(S\) is a relation \(\lt \) on \(S\) satisfying the following two axioms:
  1. For all \(a,b\in S\), exactly one holds: \(a=b\), \(a \lt b\), or \(a\gt b\).
  2. For all \(a,b,c\in S\), if \(a\lt b\) and \(b\lt c\) then \(a\lt c\).
An ordered field is a field \(F\) together with a total order \(\lt\) satisfying the following two axioms:
  1. For all \(a,b,c\in F\), if \(a\lt b\), then \(a+c\lt b+c\).
  2. For all \(a,b,c\in F\), if \(a\lt b\) and \(c \gt 0\), then \(a\cdot c\lt b\cdot c\).
Let \(F\) be an ordered field, and let \(a,b\in F\).
  1. \( 0 \lt 1\).
  2. If \(a \gt 1\), then \(a^2 \gt a\).
  3. If \(0\lt a \lt 1\), then \(a^2 \lt a\).
  4. If \(0\lt a\), then \(0 \lt 1/a\).
  5. If \(0\lt a \lt b\), then \(0 \lt b^{-1} \lt a^{-1}\).
Let \(F\) be an ordered field, and let \(a\in F\). The absolute value of \(a\), denoted by \(|a|\), is defined by \[ |a| = \begin{cases} \phantom{-}a, & \text{if }a \geq 0\\ -a, & \text{if } a \lt 0 \end{cases} \]
Let \(F\) be an ordered field, and let \(a,b,k\in F\) with \(k \gt 0\). Then
  1. \(-|a|\leq a \leq |a|\)
  2. \(|-a|=|a|\)
  3. \(|a|\lt k \iff -k \lt a \lt k\)
  4. \(|a| \gt k \iff a \gt k \text{ or }a \lt -k\)
  5. \(|ab| = |a|\cdot |b|\)
Let \(F\) be an ordered field, and let \(x,a,r\in F\) with \(r \gt 0\). Then \(|x -a| \lt r \iff x \in (a-r, a+r)\).
Let \(F\) be an ordered field, and let \(a,b\in F\). Then \(|a+b| \leq |a| + |b|\).
Let \(F\) be an ordered field, and let \(S\) be a nonempty subset of \(F\).
  1. An upper bound of \(S\) is an element \(M\in F\) such that \(M\geq s\) for all \(s\in S\). A lower bound of \(S\) is an element \(m\in F\) such that \(m\leq s\) for all \(s\in S\).
  2. \(S\) is bounded above if \(S\) has an upper bound. \(S\) is bounded below if \(S\) has a lower bound. \(S\) is bounded if it has both an upper and lower bound.
  3. A maximum of \(S\) is an upper bound of \(S\) which lies in \( S\). A minimum of \(S\) is a lower bound of \(S\) which lies in \( S\).
  4. A supremum of \(S\) is an element \(a\in F\) such that (1) \(a\) is an upper bound of \(S\) and (2) if \(b\) is an upper bound of \(S\) then \(a\leq b\). An infimum of \(S\) is an element \(c\in F\) such that (1) \(c\) is a lower bound of \(S\) and (2) if \(d\) is a lower bound of \(S\) then \(d\leq c\).
Let \(F\) be an ordered field. Then \(F\) is said to be complete if it satisfies the following Axiom of Completeness:
  1. Every subset of \(F\) which is bounded above has a supremum.

Note that a complete ordered field satisfies all eleven axioms F1 - F6, TO1 - TO2, OF1 - OF2, and AC.

There exists a unique complete ordered field.
The Real Numbers, denoted by \(\mathbb{R}\), are defined to be the unique complete ordered field whose existence is established by the above theorem.

The next four theorems are consequences of the Axiom of Completeness for \(\mathbb{R}\).

The set \(\mathbb{N}\) of natural numbers has no upper bound in \(\mathbb{R}\).
For any positive \(\epsilon\in \mathbb{R}\), no matter how small, there exists \(n\in\mathbb{N}\) such that \(0 \lt 1/n \lt \epsilon\).
For any two real numbers \(a, b\) with \(a \lt b\), there exists a rational number \(r\) such that \(a \lt r \lt b\).
For any two real numbers \(a, b\) with \(a \lt b\), there exists an irrational rational number \(r\) such that \(a \lt r \lt b\).
Cardinality
Sets \(A\) and \(B\) are said to have the same cardinality if there exists a bijection \(f:A \to B\). In this case we write \(A\sim B\).

Note that, in the above definition, if there exists a bijection \(f:A\to B\) , then there also exists a bijection \(g:B\to A\), namely, \(g=f^{-1}\). Thus the definition is symmetric in \(A\) and \(B\). (In fact it can be shown that \(\sim \) is an equivalence relation.)

A set \(A\) is
  1. finite if \(A=\emptyset\) or \(A \sim \{1,2,\ldots,n\}\) for some \(n\).
  2. infinite if \(A\) is not finite.
  3. countably infinite if \(A\sim \mathbb{N}\).
  4. countable if \(A\) is finite or countably infinite.
  5. uncountable if \(A\) is not countable.
\(\mathbb{N}\), \(\mathbb{Z}\), and \(\mathbb{Q}\) are countably infinite, but \(\mathbb{R}\) is uncountable.
Convergence of a Sequence

Informally, a sequence is an infinite list of real numbers \( (a_1, a_2, a_3,\ldots)\). Formally:

A sequence is a mapping (or function) from \(\mathbb{N}\) to \(\mathbb{R}\).

The mapping of this definition is \(n \mapsto a_n \), where \(a_n\) is the \(n\)-th term of the sequence. We often denote the sequence by \((a_n)\).

Let \( L \in \mathbb{R}\). A neighborhood of \(L\), denoted by \( N_{\epsilon}(L)\), is a set of the form \((L-\epsilon,L+\epsilon) = \{x\in\mathbb{R} : |x - L| \lt \epsilon\} \) for some \(\epsilon>0\).
A sequence \( (a_n) \) converges to a real number \(L\) if, for every positive real number \(\epsilon\), there exists \(N \in \mathbb{N}\) such that \[ n\geq N \implies |a_n - L| \lt \epsilon \]

In other words, \( (a_n) \) converges to \(L\) if, for every \( \epsilon \gt 0\), however small, there is a point in the sequence beyond which all of the terms lie in the \(\epsilon\)-neighborhood \(N_{\epsilon}(L)\). We sometimes express convergence of \( (a_n) \) to \(L\) symbolically by writing \( (a_n) \to L\) or, alternatively, \(\lim\limits_{n\to \infty} a_n = L\).

If a sequence converges, then its limit is unique.
A sequence \( (a_1, a_2, a_3, \ldots) \) is said to be bounded if the set of terms \( \{ a_1, a_2, a_3, \ldots\} \) is bounded.
If a sequence converges, then it is bounded.
Let \( (a_n) \), \( (b_n) \), and \( (c_n) \) be sequences such that \(a_n \leq b_n \leq c_n \) for all \(n\in \mathbb{N}\). If \( (a_n) \to L\) and \( (c_n) \to L\), then \( (b_n) \to L\).
Suppose that \( (a_n) \) and \( (b_n) \) are convergent sequences and \(c\in \mathbb{R}\). Then
  1. \(\lim (c a_n) = c \lim (a_n)\).
  2. \(\lim (a_n + b_n) = \lim (a_n) + \lim (b_n)\).
  3. \(\lim (a_n \cdot b_n) = \lim (a_n) \cdot \lim (b_n)\).
  4. \(\lim (a_n / b_n) = \lim (a_n) / \lim (b_n)\), provided \(b_n \neq 0\) for all \(n\) and \( \lim (b_n) \neq 0\).
A sequence \( (a_n) \) of real numbers is
  1. increasing if \(a_n \leq a_{n+1}\) for all \(n\in \mathbb{N}\).
  2. decreasing if \(a_n \geq a_{n+1}\) for all \(n\in \mathbb{N}\).
  3. monotone if it is increasing or decreasing.
  4. strictly increasing if \(a_n \lt a_{n+1}\) for all \(n\in \mathbb{N}\).
  5. strictly decreasing if \(a_n \gt a_{n+1}\) for all \(n\in \mathbb{N}\).
If a sequence is monotone and bounded, then it converges.
Let \( (a_n) \) be a sequence and let \( n_1 \lt n_2 \lt n_3 \lt \cdots \) be an increasing sequence of natural numbers. Then \[ ( a_{n_1}, a_{n_2}, a_{n_3}, \ldots) \] is called a subsequence of \( (a_n) \) and is denoted by \( (a_{n_k}) \).
If a sequence \( (a_n) \) converges to \(L\), then any subsequence of \( (a_n) \) converges to \(L\) as well.
Topology of \( \mathbb{R} \)
A set \( U \) of real numbers is open if, for every \( a \in U\), there exists \( \epsilon > 0 \) for which \( N_{\epsilon}(a) \subseteq U \).
  1. The union of any collection of open sets is open.
  2. The intersection of any finite collection of open sets is open.
Let \(L \in \mathbb{R}\). A deleted neighborhood of \(L\), denoted by \( N^{*}_{\epsilon}(L)\), is a set of the form \( (L-\epsilon,L+\epsilon)\setminus \{ L \} = \{x\in\mathbb{R} : 0 \lt |x - L| \lt \epsilon\} \), for some \(\epsilon>0\).
Let \(A\) be a subset of \( \mathbb{R}\). A point \( x\in \mathbb{R} \) is called a limit point of \(A\) (or accumulation point of \(A\) ) if every deleted neighborhood of \(x \) contains a point of \(A\).
Let \(A\) be a subset of \( \mathbb{R}\). A point \( x\in \mathbb{R} \) is a limit point of \(A\) if and only if \(x\) is the limit of a sequence of points in \(A \setminus \{ x\} \).
A set \(A \) of real numbers is closed if \(A^c\), the complement of \( A\) in \( \mathbb{R}\), is open.
  1. The union of any finite collection of closed sets is closed.
  2. The intersection of any collection of closed sets is closed.
A set is closed if and only if it contains all of its limit points.
Let \( A \) be a set of real numbers.
  1. The set of limit points of \( A\) is denoted by \( A' \).
  2. The closure of \( A\), denoted by \( \overline{A} \), is defined by \( \overline{A} = A \cup A' \).
A set \( A \) is closed if and only if \( A = \overline{A} \).
Let \( A \) be a set of real numbers.
  1. An open cover of \( A \) is a collection of open sets whose union contains \(A \).
  2. Given an open cover \( \mathcal{F} \) of \(A \), a finite subcover of \( \mathcal{F} \) is a finite subcollection of the open sets of \( \mathcal{F} \) whose union still contains \( A \).
A set \( A \) of real numbers is compact if every open cover of \( A \) has a finite subcover.
A set of real numbers is compact if and only if it is closed and bounded.
Topology of Metric Spaces
Let \( X \) be a nonempty set. A metric is function \( d : X\times X \to \mathbb{R} \) which satisfies the following four conditions: For all \( x, y, z \in X \),
  1. \( d(x,y) \geq 0 \).
  2. \(d(x,x) = 0 \), and \( d(x,y) = 0 \) only if \(x = y\).
  3. \( d(x,y) = d(y,x) \).
  4. \( d(x,y) \leq d(x,z) + d(z,y) \) (Triangle Inequality).
A metric space is a nonempty set \(X\) together with a metric \(d\). The metric space is denoted by \( (X,d)\).
Let \( (X,d ) \) be a metric space, let \(x\in X\), and let \( \epsilon>0 \). The \( \epsilon\)-neighborhood of \(x\), denoted by \(N_{\epsilon} (x) \), is \( \{y \in X : d(y,x) \lt \epsilon\} \).
Let \( (X,d) \) be a metric space. A set \( U \subseteq X \) is said to be open if, for every \( a \in U\), there exists \( \epsilon > 0 \) such that \( N_{\epsilon}(a) \subseteq U \). \( U\) is said to be closed if \( U^c \) is open.
Let \( (X,d) \) be a metric space.
  1. The union of any collection of open sets is open.
  2. The intersection of any finite collection of open sets is open.
  3. The union of any finite collection of closed sets is closed.
  4. The intersection of any collection of closed sets is closed.
Let \( (X,d) \) be a metric space. Any neighborhood is open.
Let \( (X,d) \) be a metric space. A set \( U\subseteq X \) is open if and only if \( U\) is the union of neighborhoods.
Let \( (X,d) \) be a metric space.
  1. A set \( A \subseteq X \) is compact if every cover of \( A \) has a finite subcover.
  2. A set \(A \subseteq X \) is bounded if there exists \(x\in X\) and \(\epsilon > 0\) such that \( A\subseteq N_{\epsilon}(x) \).
Let \( (X,d) \) be a metric space. Let \( A \subseteq X\). If \(A \) is compact, then \( A \) is closed and bounded.

We note that the converse of the above theorem does not hold in general. Thus the Heine-Borel Theorem does not hold for general metric spaces.

Let \( (X,d) \) be a metric space, and let \( A \subseteq X\) be a subset of \(X\). The restricted metric \( d_A:A\times A \to \mathbb{R} \) is defined by \(d_A (a,b) = d (a,b) \). In other words, \( d_A \) is the same metric as \(d\), just restricted to elements of \(A\).
Let \( (X,d) \) be a metric space, and let \( A \subseteq X\) be a subset of \(X\). Then the restricted metric \( d_A \) is a metric on \(A \). Thus \( (A, d_A) \) is a metric space.
Let \( (X,d) \) be a metric space, and let \( A \subseteq X\) be a subset of \(X\). Then the metric space \( (A, d_A ) \) is called a subspace of \( (X,d) \).
Limit of a Function
Let \(f: D\to \mathbb{R} \), where \(D\subseteq \mathbb{R} \), and let \(a \) be a limit point of \( D \). We say that \( L\in \mathbb{R} \) is a limit of \(f\) at \(a \), and write \( \lim\limits_{x\to a} f(x) = L \), if for every \(\epsilon > 0\), there exists \( \delta > 0 \) such that \[0 \lt | x-a | \lt \delta \ (\text{and }x\in D) \implies \ |f(x) - L| \lt \epsilon\]

In other words, \( \lim\limits_{x\to a} f(x) = L \), if, for every \( \epsilon \gt 0\), however small, there exists \( \delta \gt 0 \) such that all points in the in the domain of \(f \) which lie in the deleted \(\delta\)-neighborhood \( N_{\delta}^{*} (x) \) map under \( f \) to the \( \epsilon \)-neighborhood \( N_{\epsilon} (L) \).

Let \( f: D\to \mathbb{R} \) and \( g: D\to \mathbb{R} \), where \( D \subseteq \mathbb{R} \), and let \(a\) be a limit point of \(D \). Let \( k\in \mathbb{R} \).Then
  1. \(\lim c f(x) = c \lim f(x) \).
  2. \(\lim (f(x) + g(x)) = \lim f(x) + \lim g(x)\).
  3. \(\lim (f(x)\cdot g(x)) = \lim f(x) \cdot \lim g(x)\).
  4. \(\lim (f(x) / g(x)) = \lim f(x) / \lim g(x)\), provided \(g(x) \neq 0\) for all \(x\in D\) and \( \lim g(x) \neq 0\).
Let \( f: D\to \mathbb{R} \), where \( D \subseteq \mathbb{R} \), and let \(a\) be a limit point of \(D \). Let \(L \in \mathbb{R} \). Then the following are equivalent:
  1. \( \lim\limits_{x\to a} f(x) = L \).
  2. For every sequence \( (x_n) \) in \( D \setminus \{a \} \) such that \( (x_n) \to a \), we have \( f(x_n) \to L\).
Continuity
Let \(f: D\to \mathbb{R} \), where \(D\subseteq \mathbb{R} \), and let \(a\in D \). We say that \( f \) is a continuous at \(a \), if for every \(\epsilon > 0\), there exists \( \delta > 0 \) such that \[| x-a | \lt \delta \ (\text{and }x\in D) \implies \ |f(x) - f(a)| \lt \epsilon\] If \(S \) is a subset of \(D\), we say that \(f \) is continuous on \(S\) if it is continuous at all points of \( S\). We say that \(f\) is continuous if it is continuous on \(D\).
Let \(f: D\to \mathbb{R} \), where \(D\subseteq \mathbb{R} \), and let \(a\in D \).
  1. If \(a \) is a limit point of \(D \) (i.e., \(a \) is not an isolated point of \(D \)), then \(f\) is continuous at \(a \) if and only if \(\lim\limits_{x\to a} f(x) = f(a) \).
  2. If \(a \) is not a limit point of \(D \) (i.e., \(a \) is an isolated point of \(D \)), then \(f\) is continuous at \(a \).
Let \(f, g: D\to \mathbb{R} \), where \(D\subseteq \mathbb{R} \), and let \(a\in D \). Suppose that \(f\) and \(g\) are both continuous at \(a\). Then \( f+g\) and \(f\cdot g\) are continuous at \(a\), and \(f/g \) is as well assuming \(g(a)\neq 0\).
Let \(p\) be a polynomial. Then \( p\) is continuous on \( \mathbb{R} \).

We next generalize the definition of continuity to any metric space.

Suppose that \( X \) and \(Y \) are metric spaces. Let \(f:X\to Y \), and let \(a \in X\). We say that \( f \) is a continuous at \(a \), if for every neighborhood \(V\) of \(f(a) \), there exists a neighborhood \(U\) of \(a\) such that \(U \subseteq f^{-1}(V) \). If \(S \) is a subset of \(X\), we say that \(f \) is continuous on \(S\) if it is continuous at all points of \( S\). We say that \(f\) is continuous if it is continuous on \(X\).
Let \(X\) and \(Y\) be metric spaces, and let \(f:X\to Y\). The following are equivalent.
  1. \(f \) is continuous.
  2. The preimage of every open set (in \(Y \)) is open (in \(X \)).
  3. The preimage of every closed set (in \(Y \)) is closed (in \(X \)).
Let \(X\), \(Y\), and \(Z \) be metric spaces, and let \(g:X\to Y \) and \(f:Y \to Z\) be continuous. Then \(f\circ g\) is continuous.
Let \(X\) and \(Y\) be metric spaces, let \(f:X\to Y\) be continuous, and let \(A \subseteq X \). If \(A\) is compact, then \( f(A) \) is compact.
Let \(X\) be a metric space, let \(f:X\to \mathbb{R} \) be continuous, and let \(A \subseteq X \). If \(A \) is compact, then \( f \) assumes maximum and minimum values on \( A \).
Let \(f: [a,b]\to \mathbb{R} \) be continuous. Then \( f \) assumes maximum and minimum values on \( [a,b] \).
Let \( X \) be a metric space, and let \( A \subseteq X \). A separation of \(A \) is a pair of disjoint open subsets \(U \) and \(V \) of \(X \) such that
  1. \(A \subseteq U \cup V\)
  2. \( A \cap U \neq \emptyset \)
  3. \( A \cap V \neq \emptyset \)
The subset \( A \) is connected if there does not exist a separation of \(A \).
In \(\mathbb{R}\), any interval is connected.
Let \(X\) and \(Y\) be metric spaces, let \(f:X\to Y\) be continuous, and let \(A \subseteq X \). If \(A\) is connected, then \( f(A) \) is connected.
Let \(X\) be a metric space, let \(f:X\to \mathbb{R} \) be continuous, let \(A \subseteq X \), and let \( a,b\in A \). If \(A \) is connected and \( L\) is a real number lying between \( f(a) \) and \( f(b) \) (i.e., \(f(a) \lt L \lt f(b) \) or \(f(b) \lt L \lt f(a) \)), then there exists a point \( c\in A\) such that \( f(c)=L \).
Let \(f: [a,b]\to \mathbb{R} \) be continuous. If \( L\) is a real number lying between \( f(a) \) and \( f(b) \) (i.e., \(f(a) \lt L \lt f(b) \) or \(f(b) \lt L \lt f(a) \)), then there exists \( c\in (a,b) \) such that \( f(c)=L \).
Differentiation
Let \(f: I\to \mathbb{R} \), where \(I \) is an interval in \(\mathbb{R}\), and let \( c\in I \). We say that \( f \) is a differentiable at \(a \) if the limit \[ \lim\limits_{x \to c} \frac{f(x) - f(c)}{x-c} \] exists. In this case, the limit is called the derivative of \(f \) at \(c\) and denoted by \( f'(c) \). If \( f\) is differentiable at all points of some subset \( S \subseteq I\), then we say that \( f\) is differentiable on \(S\). If \(f \) is differentiable on \( I \), we say that \(f \) is differentiable.
Let \(f: I\to \mathbb{R} \), where \(I \) is an interval in \(\mathbb{R}\), and let \( c\in I \). If \( f\) is differentiable at \( c\), then \(f \) is continuous at \(c \).
Let \(f,g : I\to \mathbb{R} \), where \(I \) is an interval in \(\mathbb{R}\), and let \( c\in I \). Suppose that \( f\) and \( g\) are both differentiable at \( c\). Then \( f+g\), \( fg\), and \( f/g \) are differentiable at \(c \) (the quotient requires \(g(c) \neq 0 \)), and
  1. \((f+g)'(c) = f'(c) + g'(c)\).
  2. \((fg)'(c) =f'(c) g(c) + f(c) g'(c)\).
  3. \( \left(\dfrac{f}{g}\right)'(c) = \dfrac{f'(c) g(c) - f(c) g'(c)}{[g(c)]^2}\ \) if \( g(c) \neq 0\).
Let \( f:\mathbb{R} \to \mathbb{R} \) be given by \( f(x) = x^n \). Then \(f \) is differentiable and \( f'(c) = nc^{n-1} \) for all \( c\in \mathbb{R}\).
Let \(f: (a,b) \to \mathbb{R} \) be differentiable. If \(f \) attains a maximum or minimum at a point \(c \in (a,b) \), then \( f'(c) = 0 \).
Let \(f \) be continuous on \( [a, b]\) and differentiable on \( (a, b) \). If \(f(a) = f(b) \), then there exists at least one point \( c\in (a, b) \) such that \( f'(c) = 0 \).
Let \(f \) be continuous on \( [a, b]\) and differentiable on \( (a, b) \). Then there exists at least one point \( c\in (a,b) \) such that \[ f'(c) = \dfrac{f(b) - f(a)}{b-a}. \]
A function \( f\) is said to be increasing (resp. strictly increasing) on an interval \( I\) if \(x_1 \lt x_2 \) in \( I\) implies that \( f(x_1) \leq f(x_2) \) (resp. \( f(x_1) \lt f(x_2) \)).
Let \(f \) be continuous on \( [a, b]\) and differentiable on \( (a, b) \).
  1. If \(f'(x) = 0\) for all \(x \in (a,b)\), then \(f\) is constant on \( [a,b] \).
  2. If \(f'(x) \geq 0\) for all \(x \in (a,b)\), then \(f\) is increasing on \( [a,b] \).
  3. If \(f'(x) \gt 0\) for all \(x \in (a,b)\), then \(f\) is strictly increasing on \( [a,b] \).
Let \(F,G \) both be continuous on \( [a,b] \) and differentiable on \( (a,b) \). If \(F'(x) = G'(x)\) for all \(x\in (a,b) \), then there exists a constant \( C\) such that \( F = G + C \) on \( [a,b]\).
Let \(f \) be defined on an interval \( I\). An antiderivative of \(f\) is a function \(F \) on \(I \) such that \(F' = f\).
Let \(f \) be defined on an interval \( I\). If \(F\) and \(G\) are both antiderivatives of \(f\), then there exists a constant \(C\) such that \(F = G + C\) on \( I \).
Integration
Let \( [a,b] \) be a partition in \( \mathbb{R} \). A partition \(P \) of \( [a,b]\) is a finite set of points \( \{x_0, x_1, \ldots, x_n \} \) in \( [a,b]\) such that \[ a = x_0 \lt x_1 \lt \cdots \lt x_n=b\]
Let \(f \) be a bounded function on \( [a,b]\). For any partition \( P = \{x_0, x_1, \ldots, x_n \} \) of \( [a,b] \), define
  • \( \Delta x_i = x_i - x_{i-1}\), for \(i = 1,\ldots,n \)
  • \(M_i = \operatorname{Sup}\, \{f(x) : x\in [x_{i-1},x_i]\}\), for \(i = 1,\ldots,n\)
  • \(m_i = \operatorname{Inf}\, \{f(x) : x\in [x_{i-1},x_i]\}\), for \(i = 1,\ldots,n\)
  • \(U(f,P) = \sum\limits_{i=1}^n M_i \Delta x_i \) (upper Darbeaux sum)
  • \(L(f,P) = \sum\limits_{i=1}^n m_i \Delta x_i \) (lower Darbeaux sum)
Let \(f \) be a bounded function on \( [a,b]\). Define the upper and lower integrals of \(f\) on \([a,b]\) as follows:
  • \(\displaystyle\overline{\int_a^b} f = \operatorname{Inf} \{U(f,P) : P \text{ a partition of } [a,b] \} \)
  • \(\displaystyle\underline{\int_a^b} f = \operatorname{Sup} \{L(f,P) : P \text{ a partition of } [a,b] \} \)
Let \(f \) be a bounded function on \( [a,b]\). If the upper and lower integrals of \(f\) on \( [a,b] \) are equal, then we say that \(f\) is Riemann integrable on \([a,b]\). In this case, we denote the common value of the upper and lower integrals by \( \int_a^b f\). In other words, \[ \int_a^b f := \overline{\int_a^b} f = \underline{\int_a^b} f\] This common value is called the Riemann integral of \(f\) on \([a,b]\).