Chapter 04 Functional Limits and Continuity

4.2 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 $\lim_{x \to c} f(x) = L$ 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 $|f(x) - L| < \epsilon$.

Definition 4.2.1B (Functional Limit: Topological Version). Let $f : A \rightarrow \mathbb{R}$, and let $c$ be a limit point of the domain $A$. We say that $\lim_{x \to c} f(x) = L$ provided that, for every $\epsilon$-neighborhood $V_{\epsilon}(L)$, there exists a $\delta$-neighborhood $V_{\delta}(c)$ around $c$ with the property that for all $x \in V_{\delta}(c)$ different from $c$ with $x \in A$ 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:

(i) $\lim_{x \to c} f(x) = L$

(ii) For all sequences $(x_n) \subseteq A$ satisfying $x_n \neq c$ and $(x_n) \rightarrow c$, it follows $f(x_n) \rightarrow L$.

Summry of proof:

• $\Rightarrow$: Given $(x_n) \rightarrow c$, and $\epsilon > 0$. Since $\lim_{x \to c} f(x) = L$, we can find $V_{\delta}(c)$, as long as $x \in V_{\delta}(c)$, $|f(x) - L| < \epsilon$. Since $(x_n) \rightarrow c$, we can find $N$, when $n > N$, $x_n \in V_{\delta}(c)$. So $|f(x_n) - L| < \epsilon$. Then $f(x_n) \rightarrow L$.
• $\Leftarrow$: Use contradiction. Assume ii holds, but $\lim_{x \to c} f(x) \ne L$. Then we can find an $\epsilon$, given any $\delta$, we can find $x_{\delta}$, such that $f(x_{\delta}) \notin V_{\epsilon}(L)$. Now let $\delta_n = 1/n$. Then we can construct $(x_n) \rightarrow c$, and $f(x_n) \not\rightarrow L$, so we have an contradiction.

Corollary 4.2.4 (Algebraic Limit Theorem for Functional Limits).

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 $(x_n)$ and $(y_n)$ in $A$ with $x_n \neq c$ and $y_n \neq c$ and $\lim x_n = \lim y_n = c$ but $\lim f(x_n) \neq \lim f(y_n)$, then we can conclude that functional limit $\lim_{x \to c} f(x)$ does not exist.

4.3 Continuous Functions

Definition 4.3.1 (Continuity). A function $f: A \rightarrow \mathbb{R}$ is continuous at a point $c \in A$ if for all $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x-c| < \delta$ (and $x \in A$) it follows that $|f(x) - f(c)| < \epsilon$.

If f is continuous at every point in the domain $A$, then we say that $f$ is continuous on $A$.

Theorem 4.3.2 (Characterizations of Continuity). Let $f: A \rightarrow \mathbb{R}$ and $c \in A$. A function $f$ is continuous at a point $c$ if and only if any one of the following three conditions is met:

i. for all $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x-c| < \delta$ (and $x \in A$) it follows that $|f(x) - f(c)| < \epsilon$.

ii. for all $V_{\epsilon}(f(c))$, there exists a $V_{\delta}(c)$ such that $x \in V_{\delta}(c)$ and $x \in A$ then $f(x) \in V_{\epsilon}(f(c))$.

iii. For all $(x_n) \rightarrow c$ with $x_n \in A$, it follows $f(x_n) \rightarrow f(c)$.

If $c$ is a limit point of $A$, then the above conditions are equivalent to

iv. $\lim_{x \to c} f(x) = f(c)$.

Summary of proof:

• The equivalence of (i) and (iii). Use similar argument in theorem 4.2.3.

Corollary 4.3.3 (Criterion for Discontinuity). Let $f: A \rightarrow \mathbb{R}$ and $c \in A$ 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 $f(x)$, we may conclude that $f$ is not continuous at $c$.

Theorem 4.3.4 (Algebraic Continuity Theorem).

• $kf(x)$ is continuous at $c$ for all $k ∈ R$;
• $f(x) + g(x)$ is continuous at $c$.
• $f(x)g(x)$
• $f(x)/g(x)$

Example 4.3.5. All polynomials are continuous on $\mathbb{R}$.

Example 4.3.6.

$$g(x) = \begin{cases} x \sin (1/x) &\text{if } x \neq 0\\ 0 &\text{if } x = 0\\ \end{cases}$$

• To see $g(x)$ is continuous at 0, we can see $|g(x) - 0| = |x \sin (1/x)| \leq |x|$. So given $\epsilon$, we just need to set $\delta = \epsilon$.

Example 4.3.7. The greatest integer function $h(x) = [[x]]$.

• When $m \in \mathbf{Z}$, consider the sequence $(x_n) = m-1/n$. Then $(x_n) \rightarrow m-1 \neq m = h(m)$.
• When $c \not\in \mathbf{Z}$, let $\delta = \min \left\{ c - n, (n+1) - c \right\}$, then $h(x) = h(c) = n$.

Theorem 4.3.9 (Composition of Continuous Functions). Given $f : A→R$ and $g : B → R$, assume that the range $f(A) = {f(x) : x ∈ A}$ is contained in the domain $B$ so that the composition $g ◦ f(x) = g(f(x))$ is defined on $A$.

If $f$ is continuous at $c ∈ A$, and if $g$ is continuous at $f(c) ∈ B$, then $g ◦ f$ is continuous at $c$.

4.4 Continuous Functions on Compact Sets

Theorem 4.4.1 (Preservation of Compact Sets). Let $f : A → R$ be continuous on $A$. If $K ⊆ A$ is compact, then $f(K)$ is compact as well.

• Proof: Assume $(y_n)$ is a sequence in $f(K)$. We can find $(x_n)$ such that $f(x_n) = y_n$. Since $K$ is compact, there is a subsequence $(x_{n_k}) \rightarrow x$. Because $f$ is continuous at $x$, then $\lim_{n_k \to \infty} y_{n_k} = f(x) \in f(K)$. So we find subsequence $(y_{k_n}) \rightarrow f(x)$. So $f(K)$ is compact.

Theorem 4.4.2 (Extreme Value Theorem). If $f : K → R$ is continuous on a compact set $K ⊆ R$, then $f$ attains a maximum and minimum value. In other words, there exist $x_0,x_1 ∈ K$ such that $f(x_0) ≤ f(x) ≤ f(x_1)$ for all $x ∈ K$.

Definition 4.4.4 (Uniform Continuity). A function $f : A → R$ is uniformly continuous on $A$ if for every $ε > 0$ there exists a $δ > 0$ such that for all $x, y ∈ A$, $|x−y|<δ$ implies $|f(x)−f(y)|<ε$.

Theorem 4.4.5 (Sequential Criterion for Absence of Uniform Continuity). A function $f : A → R$ fails to be uniformly continuous on $A$ if and only if there exists a particular $ε_0 > 0$ and two sequences $(x_n)$ and $(y_n)$ in A satisfying $|x_n − y_n|→0$ but $|f(x_n)−f(y_n)|≥ε_0$.

4.5 The Intermediate Value Theorem

Theorem 4.5.1 (Intermediate Value Theorem). Let $f : [a,b] → R$ 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 $c ∈ (a,b)$ where $f(c) = L$.

Theorem 4.5.2 (Preservation of Connected Sets). Let f : G → R be continuous. If E ⊆ G is connected, then f(E) is connected as well.

Proof: Let $f(E) = A ∪ B$ where $A$ and $B$ are disjoint and nonempty.

Let

$$C=\{x∈E:f(x)∈A\} \text{ and } D=\{x∈E:f(x)∈B\}$$

• Because $A, B$ are nonempty, so $C, D$ are nonempty.
• Because $A, B$ are disjoint, so $C, D$ are disjoint.
• Also $E = C \cup D$.

Since $E$ is connected, we can assume there is a sequence $(x_n) \in C$ such that $\lim x_n = x \in D$. Since $f$ is continuous, then $f(x_n) \in A$ and $\lim f(x_n) = f(x) \in B$. So $f(E)$ is also connected. $\square$

Definition 4.5.3. 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 $c ∈ (x,y)$ where $f(c) = L$.

4.6 Sets of Discontinuity

Given a function $f: \mathbf{R} → \mathbf{R}$, define $D_f ⊆R$ to be the set of points where the function $f$ fails to be continuous.

• Dirichlet’s function: $D_f = \mathbf{R}$.
• Modified Dirichlet’s function: $D_h = \mathbf{R} \backslash \{0\}$.
• Thomae function: $D_t = \mathbf{Q}$.

Definition 4.6.2. Given a limit point $c$ of a set $A$ and a function $f : A → R$, we write

$$\lim_{x \to c^+} f(x) = L$$

if for all $ε > 0$ there exists a $δ > 0$ such that $|f(x)−L| < ε$ whenever $0 < x−c < δ$.

Equivalently, in terms of sequences, $\lim_{x \to c^+} f(x) = L$ if $\lim f(x_n) = L$ for all sequences $(x_n)$ satisfying $x_n > c$ and $\lim (x_n) = c$.

Theorem 4.6.3. Given $f : A → R$ and a limit point $c$ of $A$, $\lim_{x \to c} f(x) = L$ if and only if

$$\lim_{x \to c^+} f(x) = L \text{ and } \lim_{x \to c^-} f(x) = L.$$

Generally speaking, discontinuities can be divided into three categories:

• If $\lim_{x \to c}f(x)$ exists but has a value different from $f(c)$, the discontinuity at $c$ is called removable.
• If $\lim_{x \to c^+}f(x) \not = \lim_{x \to c^-} f(x)$ then $f$ has a jump discontinuity at $c$.
• If $\lim_{x \to c}f(x)$ does not exist for some other reason, then the discontinuity at $c$ is called an essential discontinuity.

$D_f$ for an Arbitrary Function

Definition 4.6.4. A set that can be written as the countable union of closed sets is in the class $F_σ$. (This definition also appeared in Section 3.5.)

Definition 4.6.5. Let $f$ be defined on $\mathbf{R}$, and let $α > 0$. The function $f$ is $α$-continuous at $x ∈ \mathbf{R}$ if there exists a $δ > 0$ such that for all $y,z ∈ (x−δ,x+δ)$ it follows that $|f(y) − f(z)| < α$.

Given a function $f$ on $\mathbf{R}$, define $D_f^α$ to be the set of points where the function $f$ fails to be $α$-continuous. In other words,

$$D_f^α = \{ x∈ \mathbf{R}:f \text{ is not α-continuous at x} \}$$ .