Real Analysis (3)

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Real Analysis (3)

Basic Topology of \(\mathbb{R}\)

Open and Closed Sets

Definition 3.2.1: Open

A set \(O \subseteq \mathbb{R}\) is open if for all points \(a \in O\) there exists an \(\epsilon-neighborhood V_{\epsilon} (a) \subseteq O\).

\(\mathbb{R}\) is open \(\emptyset\) is open

Theorem 3.2.3

  1. The union of an arbitrary collection of open sets is open.
  2. The intersection of a finite collection of open sets is open.

Definition 3.2.4: Limit Point

A point \(x\) is a limit point of a set \(A\) if every \(\epsilon-neighborhood V_{\epsilon} (x)\) of \(x\) intersects the set \(A\) at some point other than \(x\). Limit points may not be in \(A\), consider the end points of open sets.

Theorem 3.2.5

A point \(x\) is a limit point of a set \(A\) if and only if \(x = \lim_{n \rightarrow \infty} a_n\) for some sequence \((a_n)\) contained in \(A\) satisfying \(a_n \neq x, \; \forall n \in \mathbb{N}\)

Definition 3.2.6: Isolated Point

A point \(a \in A\) is an isolated point of \(A\) if it is not a limit point of \(A\). Isolated point is always in \(A\).

Definition 3.2.7: Closed Set

A set \(F \subseteq \mathbb{R}\) is closed if it contains all its limit points. Topologically speaking, a closed set is one where convergent sequences within the set have limits that are also in the set.

Theorem 3.2.8

A set \(F \subseteq \mathbb{R}\) is closed if and only if every Cauchy sequence contained in \(\mathbb{F}\) has a limit that is also an element of \(F\)

Theorem 3.2.10: Density of \(\mathbb{Q}\) in \(\mathbb{R}\)

For every \(y \in \mathbb{R}\), there exists a sequence of rational numbers that converges to \(y\).

Definition 3.2.11: Closure

Given a set \(A \subseteq \mathbb{R}\), let \(L\) be the set of all limit points of \(A\). The closure of \(A\) is defined to be \(\bar{A} = A \cup L\).

That is, if \(A\) is an open interval \((a, b)\), then \(\bar{A} = [a, b]\). If \(A\) is a closed interval, then \(\{\bar{A} = A\}\). \(\bar{A}\) is always a closed set.

Theorem 3.2.12

For any \(A \subseteq \mathbb{R}\), the closure \(\bar{A}\) is a closed set and is the smallest closed set containing \(A\).

Complement

In general, if a set is not open that does not imply it must be closed. Many sets such as half-open interval \((c,d]\) are neither open nor closed. \(\emptyset, \mathbb{R}\) are both open and closed. The complement of a set \(A \subseteq \mathbb{R}\) is defined to be the set:

\[A^c = \{x \in \mathbb{R}: x \notin A\}\]

Theorem 3.2.13

A set \(O\) is open if and only if \(O^c\) is closed. Likewise, a set \(F\) is closed if and only if \(F^c\) is open.

Theorem 3.2.14

  1. The union of a finite collection of closed sets is closed.
  2. The intersection of an arbitrary collection of closed sets is closed.

Compact Sets

Definition 3.3.1: Compactness

A set \(K \subseteq \mathbb{R}\) is compact if every sequence in \(K\) has a subsequence that converges to a limit that is also in \(K\).

Theorem 3.3.3 Bounded Sets

A set \(A \subseteq \mathbb{R}\) is bounded if there exists \(M > 0\) such that \(|a| \leq M, \;\forall a \in A\)

Theorem 3.3.4: Characterization of Compactness in \(\mathbb{R}\)

A set \(K \subseteq \mathbb{R}\) is compact if and only if it is closed and bounded.

We can think of compact sets as generalizations of closed intervals. Whenever a fact involving closed intervals is true, it is often the case that the same results holds when we replace closed interval with compact set.

Theorem 3.3.5: Nested Compact Set Property

If \(K_1 \subseteq K_2 \subseteq K_3 \subseteq ..\) is a nested sequence of nonempty compact sets, then the intersection \(bigcap^{\infty}_{n=1} K_n\) is nonempty.

Perfect Sets and Connected Sets

Definition 3.4.1

A set \(P \subseteq \mathbb{R}\) is perfect if it is closed and contains no isolated points (eg. closed intervals on real numbers)

Theorem 3.4.3

A nonempty perfect set is uncountable.

Definition 3.5.1

A set \(A \subseteq \mathbb{R}\) is called an \(F_\sigma\) set if it can be written as the countable union of closed sets. A set \(B \subseteq \mathbb{R}\) is called a \(G_\delta\) set if it can be written as the countable intersection of open sets.

Theorem 3.5.2

If \(\{G_1, ...\}\) is a countable collection of dense, open sets, then the intersection \(\bigcap^{\infty}_{n=1} G_n\) is not empty.

Functional Limits and Continuity

Functional Limits

Definition 4.2.1: Functional Limit

Let \(f: A \rightarrow \mathbb{R}\), and let \(c\) be a limit point of the domain \(A\). We say that \(\lim_{x \rightarrow c} f(x) = L\), provided that, for all \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < | x - c | < \delta\) and \(x \in A\) it follows that \(| f(x) - L | < \epsilon\). In other words, Let \(c\) be a limit point of the domain of \(f: A \rightarrow \mathbb{R}\). We say \(\lim_{x \rightarrow c} f(x) = L\) provided that, for every \(V_{\epsilon} (L)\), there exists a \(V_{\delta} (c)\) with the property that for all \(x \in V_{\delta} (c)\) different from \(c\), it follows that \(f(x ) \in V_{\epsilon} (L)\).

Theorem 4.2.3: Sequential Criterion for Functional Limits

Given a function \(f: A \rightarrow \mathbb{R}\) and a limit point \(c\) of \(A\), the following two statements are equivalent:

  1. \(\lim_{x \rightarrow c} f(x) = L\)
  2. For all sequences \((x_n) \subseteq A\) satisfying \(x_n \neq c\) and \((x_n) \rightarrow c\), it follows that \(f(x_n) \rightarrow L\).

Corollary 4.2.4: Algebraic Limit Theorem for Functional Limits

Let \(f, g\) be functions defined on a domain \(A \subseteq \mathbb{R}\), and assume \(\lim_{x \rightarrow c} f(x) = L\) and \(\lim_{x \rightarrow c} g(x) = M\), for some limit point \(c\) of \(A\). Then,

  1. \(\lim_{x \rightarrow c} kf(x) = kL\)
  2. \(\lim_{x \rightarrow c} [f(x) + g(x)] = L + M\)
  3. \(\lim_{x \rightarrow c} f(x)g(x) = LM\)
  4. \(\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{L}{M}\), given \(M \neq 0\)

Corollary 4.2.5: Divergence Criterion for Functional Limits

Let \(f\) be a function defined on \(A\), and let \(c\) be a limit point of \(A\). If there exist two sequences \((x_n), (y_n)\) in \(A\) with \(x_n \neq c\) and \(y_n \neq c\) and

\[\lim_{n \rightarrow \infty} x_n = \lim_{n \rightarrow \infty} y_n = c\]

but

\[\lim_{n \rightarrow \infty} f(x_n) \neq \lim_{n \rightarrow \infty} f(y_n) \]

then, we can conclude that the functional limit \(\lim_{n \rightarrow c} f(x)\) does not exist.

Continues Functions

Definition 4.3.1: Continuity

A function \(f: A \rightarrow \mathbb{R}\) is continues at a point \(c \in A\) if for all \(\epsilon > 0\), there exists a \(\delta > 0\) s.t whenever \(|x - c| < \delta\) it follows that \(|f(x) - f(c)| < \epsilon\).

If \(c\) is a limit point of \(A\), we can reduce the definition to say that, \(f\) is continuous at \(c \in A\) if

\[\lim_{x \rightarrow c} f(x) = f(c)\]

If \(f\) is continues at every point in the domain \(A\), then we say that \(f\) is continues on \(A\).

Theorem 4.3.2: Characterizations of Continuity

Let \(f: A \rightarrow \mathbb{R}\) and let \(c \in A\). The function \(f\) is continuous at \(c\) if and only if any one of the following three conditions is met:

  1. For all \(\epsilon > 0\), there exists a \(\delta > 0\) s.t \(|x - c| < \delta\) implies \(|f(x) - f(c)| < \epsilon\).
  2. For all \(V_{\epsilon} (f(c))\), there exists a \(V_{\delta} (c)\) with the property that \(x \in V \implies f(x) \in V_{\epsilon} (f(c))\).
  3. For all \((x_n) \rightarrow c\), it follows that \(f(x) \rightarrow f(c)\).
  4. If \(c\) is a limit point of \(A\), then the above conditions equivalent to \[\lim_{x \rightarrow c} f(x) = f(c)\]

Corollary 4.3.3: Criterion for Discontinuity

Let \(f: A \rightarrow \mathbb{R}\), and let \(c \in A\) be a limit point of \(A\). If there exists a sequence \((x_n) \subseteq A\) where \((x_n) \rightarrow c\), but such that \(f(x_n)\) does not converge to \(f(c)\), we may conclude that \(f\) is not continues at \(c\).

Theorem 4.3.4: Algebraic Continuity Theorem

Assume \(f: A \rightarrow \mathbb{R}\) and \(g: A \rightarrow \mathbb{R}\) are continuous at a point \(c \in A\). Then,

  1. \(kf(x)\) is continuous at \(c\) for all \(k \in \mathbb{R}\).
  2. \(f(x) + g(x)\) is continuous at \(c\).
  3. \(f(x) g(x)\) is continuous at \(c\).
  4. \(\frac{f(x)}{g(x)}\) is continuous at c, provided the quotient is defined.

Theorem 4.3.9: Composition of Continuous Functions

Given \(f: A \rightarrow \mathbb{R}\) and \(b \rightarrow \mathbb{R}\), assume that the range \(f(A) = \{f(x): x \in A\}\) is contained in the domain \(B\) so that the composition \(g \circ f(x) = g(f(x))\) is defined on \(A\).

If \(f\) is continuous at \(c \in A\), and if \(g\) is continuous at \(f(c) \in B\), then \(g \circ f\) is continuous at \(c\).

Continuous Functions on Compact Sets

Given a function \(f: A \rightarrow \mathbb{R}\) and a subset \(B \subseteq A\), the notation \(f(B)\) refers to the range of \(f\) over the set \(B\), that is,

\[f(B) = \{f(x): x \in B\}\]

Theorem 4.4.1: Preservation of Compact Sets

Let \(f: A \rightarrow \mathbb{R}\) be continuous on \(A\), if \(K \subseteq A\) is compact, then \(f(K)\) is compact as well.

Theorem 4.4.2: Extreme Value Theorem

If \(f: K \rightarrow \mathbb{R}\) is continuous on a compact set \(K \subseteq \mathbb{R}\), then \(f\) attains a maximum and minimum value. In other words, there exist \(x_0, x_1 \in K\) such that \(f(x_0) \leq f(x) \leq f(x_1)\) for all \(x \in K\).

Definition 4.4.4: Uniform Continuity (stronger than continuity)

A function \(f: A \rightarrow \mathbb{R}\) is uniformly continuous on \(A\) if for every \(\epsilon > 0\) there exists a \(\delta > 0\) such that for all \(x, y \in A\), \(|x - y| < \delta \implies |f(x) - f(y)| < \epsilon\).

The difference between saying "f is continuous on A" and "f is uniformly continuous on A" is that, for the first definition, given \(\epsilon > 0\) and \(c \in A\), we can find a \(\delta > 0\) s.t the conditions are satisfied. However, for a function to be uniformly continuous, given \(\epsilon > 0\), we need to find a \(\delta > 0\) that works for all \(c \in A\). Thus, uniform continuity is stronger.

Theorem 4.4.5: Sequential Criterion for Absence of Uniform Continuity

A function \(f: A \rightarrow \mathbb{R}\) fails to be uniformly continuous on \(A\) if and only if there exists a particular \(\epsilon_0 > 0\) and two sequences \((x_n), (y_n)\) in \(A\) satisfying:

\[|x_n - y_n| \rightarrow 0\]

but

\[|f(x_n) - f(y_n)| \geq \epsilon_0\]

Theorem 4.4.7: Uniform Continuity on Compact Sets

A function that is continuous on a compact set \(K\) is uniformly continuous on \(K\).

Theorem 4.5.1: Intermediate Value Theorem

Let \(f: [a, b] \rightarrow \mathbb{R}\) be continuous. If \(L\) is a real number satisfying \(f(a) < L < f(b)\) or \(f(a) > L > f(b)\), then there exists a point \(c \subseteq (a, b)\) where \(f(c) = L\).

Definition 4.5.3: Intermediate Value Property

A function \(f\) has the intermediate value property on an interval \([a, b]\) if for all \(x < y\) in \([a, b]\) and all \(L\) between \(f(x)\) and \(f(y)\), it is always possible to find a point \(c \in (x, y)\) where \(f(c) = L\).

Another way to summarize the intermediate value theorem is to day that every continuous function on \([a, b]\) has the intermediate value property.

Sets of Discontinuity

Definition 4.6.1: Monotonic Function

A function \(f: A \rightarrow \mathbb{R}\) is increasing on \(A\) if \(f(x) \leq f(y)\) whenever \(x < y\) and decreasing if \(f(x) \geq f(y)\) whenever \(x < y\) in \(A\). A monotone function is one that is either increasing or decreasing.

Definition 4.6.2 Right Hand Limit

Given a limit point \(c\) of a set \(A\) and a function \(f: A \rightarrow \mathbb{R}\), we write:

\[\lim_{x \rightarrow c^+} f(x) = L\]

if for all \(\epsilon > 0\) there exists a \(\delta > 0\) s.t \(|f(x) - L| < \epsilon\) whenever \(0 < x - c < \delta\).

Definition 4.6.3 Left Hand Limit

Given a limit point \(c\) of a set \(A\) and a function \(f: A \rightarrow \mathbb{R}\), we write:

\[\lim_{x \rightarrow c^-} f(x) = L\]

if for all \(\epsilon > 0\) there exists a \(\delta > 0\) s.t \(|f(x) - L| < \epsilon\) whenever \(-\delta < x - c < 0\).

Theorem 4.6.4

Given \(f: A \rightarrow \mathbb{R}\) and a limit point \(c\) of \(A\), \(lim_{x \rightarrow c} f(x) = L\) if and only if

\[\lim_{x \rightarrow c^-} f(x) = L\]

and

\[\lim{x \rightarrow c^+} f(x) = L\]

Definition 4.6.5: \(\alpha\)-continuous

Let \(f\) be defined on \(\mathbb{R}\), and let \(\alpha > 0\). The function \(f\) is \(\alpha\)-continuous at \(x \in \mathbb{R}\) if there exists a \(\delta > 0\) s.t for all \(y, z \in (x - \delta, x + \delta)\) it follows that \(f(y) - f(z)| < \alpha\).

Derivatives

Derivatives and the Intermediate Value Property

Definition 5.2.1: Differentiability

Let \(g: A \rightarrow \mathbb{R}\) be a function defined on an interval \(A\). Given \(c \in A\), the derivative of \(g\) at \(c\) is defined by:

\[g^{\prime} (c) = \lim_{x \rightarrow c} \frac{g(x) - g(c)}{x - c}\]

provided this limit exists. In this case, we say \(g\) is differentiable at \(c\). If \(g^{\prime}\) exists for all points \(c \in A\), we say that \(g\) is differentiable on \(A\).

Theorem 5.2.3

If \(g: A \rightarrow \mathbb{R}\) is differentiable at a point \(c \in A\), then \(g\) is continuous at \(c\) as well.

Theorem 5.2.4: Algebraic Differentiability Theorem

Let \(f\) and \(g\) be functions defined on an interval \(A\), and assume both are differentiable at some point \(c \in A\). Then,

  1. \((f + g)^{\prime} (c) = f^{\prime} (c) + g^{\prime} (c)\)
  2. \((kf)^{\prime} (c) = kf^{\prime} (c), \forall k \in \mathbb{R}\)
  3. \((fg)^{\prime} (c) = f^{\prime} (c) g(c) + f(c) g^{\prime} (c)\)
  4. \((\frac{f}{g})^{\prime} (c) = \frac{g(c)f^{\prime}(c) - f(c)g^{\prime}(c)}{[g(c)]^2}, \;\; g(c) \neq 0\)

Theorem 5.2.5: Chain Rule

Let \(f: A \rightarrow \mathbb{R}\) and \(g: B \rightarrow \mathbb{R}\) satisfy \(f(A) \subseteq B\) so that the composition \((g \circ f)^{\prime}\) is defined. If \(f\) is differentiable at \(c \in A\) and if \(g\) is differentiable at \(f(c) \in B\), then \(g \circ f\) is differentiable at \(c\) with:

\[(g\circ f)^{\prime}(c) = g^{\prime} (f(c)) \cdot f^{\prime} (c)\]

Theorem 5.2.6: Interior Extremum Theorem

Let \(f\) be differentiable on an open interval \((a, b)\). If \(f\) attains a maximum value at some point \(c \in (a, b)\) (i.e \(f(c) \geq f(x) \; \forall x \in (a, b)\)), then \(f^{\prime}(c) = 0\). The same is true if \(f(c)\) is a minimum value.

Theorem 5.2.7: Darboux's Theorem

If \(f\) is differentiable on an interval \([a, b]\), and if \(\alpha\) satisfies \(f^{\prime}(a) < \alpha < f^{\prime} (b)\) or \(f^{\prime} > \alpha > f^{\prime} (b)\), then there exists a point \(c \in (a, b)\) where \(f^{\prime} (c) = \alpha\).

Mean Value Theorem

Theorem 5.3.1: Rolle's Theorem

Let \(f: [a, b] \rightarrow \mathbb{R}\) be continuous on \([a, b]\) and differentiable on \((a, b)\). If \(f(a) = f(b)\), then there exists a point \(c \in (a, b)\) where \(f^{\prime} (c) = 0\).

Theorem 5.3.2: Mean Value Theorem

If \(f: [a, b] \rightarrow \mathbb{R}\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists a point \(c \in (a, b)\) where

\[f^{\prime} (c) = \frac{f(b) - f(a)}{b - a}\]

Corollary 5.3.3

If \(g: A \rightarrow \mathbb{R}\) is differentiable on an interval \(A\) and satisfies \(g^{\prime}(x) = 0, \;\forall x \in A\), then \(g(x) = k\) for some constant \(k \in \mathbb{R}\).

Corollary 5.3.4

If \(f, g\) are differentiable functions on an interval \(A\) and satisfy \(f^{\prime} (x) = g^{\prime} (x), \; \forall x \in A\), then \(f(x) = g(x) + k\) for some constant \(k \in \mathbb{R}\).

Theorem 5.3.5: Generalized Mean Value Theorem

If \(f\) and \(g\) are continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists a point \(c \in (a, b)\) where

\[[f(b) - f(a)]g^{\prime} (c) = [g(b) - g(a)] f^{\prime} (c)\]

If \(g^{\prime}\) is never zero on \((a, b)\), then the conclusion can be stated as:

\[\frac{f^{\prime}(c)}{g^{\prime}(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}\]

Theorem 5.3.6: L’Hospital’s Rule: 0/0 case

Let \(f\) and \(g\) be continuous on an interval containing \(a\), and assume \(f\) and \(g\) are differentiable on this interval with the possible exception of the point \(a\). If \(f(a) = g(a) = 0\) and \(g^{\prime} (x) \neq 0, \; \forall x \neq a\), then:

\[\lim_{x \rightarrow a} \frac{f^{\prime} (x)}{g^{\prime} (x)} = L \implies \lim_{x \rightarrow a} \frac{f (x)}{g (x)} = L\]

Definition 5.3.7

Given \(g: A \rightarrow \mathbb{R}\) and a limit point \(c \in A\), we say that \(\lim_{x \rightarrow c} g(x) = \infty\) if, for every \(M > 0\), there exists a \(\delta > 0\) such that whenever \(0 < | x - c | < \delta\) it follows that \(g(x) \geq M\). We can define \(\lim_{x \rightarrow c} g(x) = -\infty\) in a similar way.

Theorem 5.3.8: L’Hospital’s Rule: \(\infty/\infty\) case

Let \(f\) and \(g\) be differentiable on \((a, b)\). If \(\lim_{x \rightarrow a} g(x) = \infty\) or \(-\infty\) and \(g^{\prime} (x) \neq 0, \; \forall x \neq a\), then:

\[\lim_{x \rightarrow a} \frac{f^{\prime} (x)}{g^{\prime} (x)} = L \implies \lim_{x \rightarrow a} \frac{f (x)}{g (x)} = L\]

Moreover, these functions have derivatives of all orders.

Theorem

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