2 Continuity of Functions

2.1 Cardinality of Sets

Cardinality of Sets

Let $\mathbb{N}^*$ be the set of all positive integers, and

$$\begin{equation*} N_n = \{1, 2, \cdots, n\}. \end{equation*}$$

$\text{(1)}\quad$If there exists a positive integer $n$ such that set $A \sim N_n$, then $A$ is called a finite set. The empty set is also considered a finite set.

$\text{(2)}\quad$If set $A$ is not a finite set, then $A$ is called an infinite set.

$\text{(3)}\quad$If $A \sim \mathbb{N}^*$, then $A$ is called a countable set.

$\text{(4)}\quad$If $A$ is neither a finite set nor a countable set, then $A$ is called an uncountable set.

$\text{(5)}\quad$If $A$ is a finite set or $A$ is a countable set, then $A$ is called at most countable.

Theorem 2.1. Every infinite subset of a countable set $A$ is a countable set.

Theorem 2.2. Let $\{E_n\}$ ($n=1, 2, 3, \cdots$) be a sequence of at most countable sets. Let

$$\begin{equation*} S = \bigcup_{n=1}^{\infty} E_n, \end{equation*}$$

then $S$ is an at most countable set.

2.2 Elementary Functions

Constant Function

$$\begin{equation*} f(x) = c, \quad c \in \mathbb{R}. \end{equation*}$$

• Domain: $\mathbb{R}$.

• Range: $\{c \}$.

Identity Function

$$\begin{equation*} f(x) = x. \end{equation*}$$

• Domain: $\mathbb{R}$.

• Range: $\mathbb{R}$.

Exponential Function

$$\begin{equation*} f(x) = a^x, \quad a>0,\; a\neq1. \end{equation*}$$

• Domain: $\mathbb{R}$.
• Range: $(0, +\infty)$.
• Special case: $f(x) = e^x$ is the natural exponential function.

Logarithmic Function

$$\begin{equation*} f(x) = \log_a x, \quad a>0,\; a\neq1. \end{equation*}$$

• Domain: $(0, +\infty)$.
• Range: $\mathbb{R}$.
• Special case: $f(x) = \ln x = \log_e x$.
• Inverse: $(\log_a x)^{-1} = a^x.$

Power Function

$$\begin{equation*} f(x) = x^a = e ^ {a \ln x}, \quad a \in \mathbb{R}. \end{equation*}$$

• Domain: $(0, +\infty)$.
• Range: $(0, +\infty)$.
• Typical examples: $x^2,\; x^3,\; \sqrt{x},\; \dfrac{1}{x}$.

Trigonometric Function

$$\begin{equation*} \sin x,\; \cos x,\; \tan x = \frac{\sin x}{\cos x},\; \cot x = \frac{\cos x}{\sin x},\; \sec x = \frac{1}{\cos x},\; \csc x = \frac{1}{\sin x}. \end{equation*}$$

• Domain and Range:

Function	Domain	Range
$\sin x$	$\mathbb{R}$	$[-1,1]$
$\cos x$	$\mathbb{R}$	$[-1,1]$
$\tan x$	$x\neq \pi/2 + k\pi$	$\mathbb{R}$

Hyperbolic Function

$$\begin{equation*} \sinh x = \frac{e^x - e^{-x}}{2}, \qquad \cosh x = \frac{e^x + e^{-x}}{2}, \qquad \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}. \end{equation*}$$

• Domain and Range:

Function	Domain	Range
$\sinh x$	$\mathbb{R}$	$\mathbb{R}$
$\cosh x$	$\mathbb{R}$	$[1,\,+\infty)$
$\tanh x$	$\mathbb{R}$	$(-1,\,1)$

Theorems

1. $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\\$.

2. $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\\$.

3. $\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}.$

4. $\sin(2A)=2\sin A\cos A$.

5. $\cos(2A)=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A$.

6. $\tan(2A)=\frac{2\tan A}{1-\tan^2A}$.

7. $\sin A+\sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$.

8. $\sin A-\sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$.

9. $\cos A+\cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$.

10. $\cos A-\cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}$.

11. $\sin A\sin B = \tfrac{1}{2}\big[\cos(A-B)-\cos(A+B)\big]$.

12. $\cos A\cos B = \tfrac{1}{2}\big[\cos(A-B)+\cos(A+B)\big]$.

13. $\sin A\cos B = \tfrac{1}{2}\big[\sin(A+B)+\sin(A-B)\big]$.

14. For angles $\theta$ in the interval $(0,\tfrac{\pi}{2}):$ $\sin\theta < \theta < \tan\theta$.

15. $\sin(2x) = 2 \sin x \cos x$.

16. $\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$.

17. $\sinh(2x) = 2 \sinh x \cosh x$.

18. $\cosh(2x) = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 1 + 2\sinh^2 x$.

19. $\operatorname{asinh}(x) = \ln\!\left(x + \sqrt{x^2 + 1}\right)$.

20. $\operatorname{acosh}(x) = \ln\!\left(x + \sqrt{x - 1}\sqrt{x + 1}\right)$.

Inverse Trigonometric Function

$$\begin{equation*} \arcsin x,\quad \arccos x,\quad \arctan x. \end{equation*}$$

• Domain and Range:

Function	Domain	Range
$\arcsin x$	$[-1,1]$	$[-\pi/2,\ \pi/2]$
$\arccos x$	$[-1,1]$	$[0,\ \pi]$
$\arctan x$	$\mathbb{R}$	$(-\pi/2,\ \pi/2)$

Elementary Function

If $f,g$ are elementary functions, then

$$\begin{equation*} f+g,\quad f-g,\quad f\cdot g,\quad \frac{f}{g}\;(g\neq0),\quad f\circ g \end{equation*}$$

are all elementary functions (in their domains of definition).

2.3 Limit of Functions

Limits of Function

A real number $x_0$ is called an accumulation point (or limit point) of a set $I \subseteq \mathbb{R}$ if:

$$\begin{equation*} \forall \varepsilon > 0,\; \exists x \in I \text{ such that } 0 < |x - x_0| < \varepsilon. \end{equation*}$$

Let $f : I \to \mathbb{R}$ and let $x_0$ be an accumulation point of $I$. We say that the limit of $f(x)$ as $x$ approaches $x_0$ is $A \in \mathbb{R}$, written as

$$\begin{equation*} \lim_{x \to x_0} f(x) = A. \end{equation*}$$

if and only if

$$\begin{equation*} \forall \varepsilon > 0,\; \exists \delta > 0 \text{ such that } \forall x \in I,\; 0 < |x - x_0| < \delta \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$

• The right-hand limit of a function $f$ at $x_0$ is denoted by

  $$\begin{equation*} \lim_{x \to x_0^+} f(x) = A, \end{equation*}$$

  which means:

  $$\begin{equation*} \forall \varepsilon > 0,\; \exists \delta > 0 \text{ such that } \forall x \in I,\; x_0 < x < x_0 + \delta \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$

• Similarly, the left-hand limit is defined by

  $$\begin{equation*} \lim_{x \to x_0^-} f(x) = A. \end{equation*}$$

Theorem 2.3. A necessary and sufficient condition for the function $f$ to have a limit $l$ at $x_0$ is that, for any sequence $\{x_n \neq x_0 : n=1, 2, 3, \cdots\}$ converging to $x_0$, the sequence $\{f(x_n)\}$ has a limit $l$.

Theorem 2.4. Let $\lim\limits_{x \to x_0} f(x)$ and $\lim\limits_{x \to x_0} g(x)$ exist. Then:

$\text{(1)}\quad\lim\limits_{x \to x_0} (f \pm g)(x) = \lim\limits_{x \to x_0} f(x) \pm \lim\limits_{x \to x_0} g(x)$;

$\text{(2)}\quad\lim\limits_{x \to x_0} fg(x) = \lim\limits_{x \to x_0} f(x) \cdot \lim\limits_{x \to x_0} g(x)$;

$\text{(3)}\quad\lim\limits_{x \to x_0} \frac{f}{g}(x) = \frac{\lim\limits_{x \to x_0} f(x)}{\lim\limits_{x \to x_0} g(x)}$, where $\lim\limits_{x \to x_0} g(x) \neq 0$.

Neighborhood and Deleted Neighborhood

$$\begin{equation*} B_\delta(x_0) = \{x : |x - x_0| < \delta\}, \quad B_\delta(\mathring{x}_0) = \{x : 0 < |x - x_0| < \delta\}. \end{equation*}$$

We call $B_\delta(x_0)$ the neighborhood of $x_0$ centered at $x_0$ with radius $\delta$ (abbreviated as the neighborhood of $x_0$), and $B_\delta(\mathring{x}_0)$ the deleted neighborhood of $x_0$ centered at $x_0$ with radius $\delta$ (abbreviated as the deleted neighborhood of $x_0$).

Theorem 2.5. A necessary and sufficient condition for the function $f$ to have a limit at $x_0$ is that for any given $\varepsilon > 0$, there exists a $\delta > 0$ such that for any $x_1, x_2 \in B_\delta(\mathring{x}_0)$, we have $|f(x_1) - f(x_2)| < \varepsilon$.

Theorem 2.6. Let $\lim\limits_{x \to x_0} f(x) = l$ and $\lim\limits_{t \to t_0} g(t) = x_0$. If in some neighborhood $B_{\eta}(t_0)$ of $t_0$, $g(t) \neq x_0$, then

$$\begin{equation*} \lim_{t \to t_0} f(g(t)) = l. \end{equation*}$$

2.4 Infinite Limits of Functions

Limit at Infinity

$$\begin{equation*} \lim_{x \to +\infty} f(x) = A \end{equation*}$$

means that

$$\begin{equation*} \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; x > M \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$

Similarly,

$$\begin{equation*} \lim_{x \to -\infty} f(x) = A \end{equation*}$$

means

$$\begin{equation*} \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; x < -M \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$ $$\begin{equation*} \lim_{x \to \infty} f(x) = A \end{equation*}$$

means

$$\begin{equation*} \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; |x| > M \Rightarrow |f(x) - A| < \varepsilon. \end{equation*}$$

Definition. Let $c$ be one of $x_0,\; x_0^+,\; x_0^-,\; -\infty,\; +\infty,\; \infty$. The neighborhood and deleted neighborhood of $c$ are defined as follows:

$c$	Neighborhood of $c$	Deleted neighborhood of $c$
$x_0$	An open interval $J$ containing $x_0$	$J \setminus \{x_0\}$
$x_0^+$	$[x_0,\, x_0+\delta)$	$(x_0,\, x_0+\delta)$
$x_0^-$	$(x_0-\delta,\, x_0]$	$(x_0-\delta,\, x_0)$
$-\infty$	$(-\infty,\, b)$	$(-\infty,\, b)$
$+\infty$	$(a,\, +\infty)$	$(a,\, +\infty)$
$\infty$	$(-\infty,\, b)\,\cup\,(a,\, +\infty)$	$(-\infty,\, b)\,\cup\,(a,\, +\infty)$

General Definition of Limit

Let $c$ be one of $x_0,\; x_0^+,\; x_0^-,\; -\infty,\; +\infty,\; \infty$, and let $A$ be a real number or one of $-\infty,\; +\infty,\; \infty$. We say that $\lim\limits_{x \to c} f(x) = A$ iff:

For every neighborhood $V$ of $A$, there exists a deleted neighborhood $W$ of $c$ such that for all $x \in I \cap W$, we have $f(x) \in V$.

Infinite Quantity

When $x \to c$, if $f(x)$ grows without bound, we say $f(x)$ is an infinite quantity (or positive/negative infinite quantity).

If

$$\begin{equation*} \lim_{x \to c} f(x) = \infty \quad \text{(i.e. } \lim_{x \to c} f(x) = +\infty \text{ or } \lim_{x \to c} f(x) = -\infty \text{)}, \end{equation*}$$

then $f(x)$ diverges to infinity as $x$ approaches $c$.

Theorem 2.7.

$$\begin{equation*} \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e. \end{equation*}$$

2.5 Big O and Small o

Big O

When $x \to c$, if $f(x) = O(g(x))$, it means:

There exists $M > 0$ and a deleted neighborhood $W$ of $c$ such that for all $x \in I \cap W$, we have $|f(x)| \le M |g(x)|$.

• If $f_1 = O(g)$ and $f_2 = O(g)$ as $x \to c$, then $f_1 + f_2 = O(g).$

• If $f_1 = O(g_1)$ and $f_2 = O(g_2)$ as $x \to c$, then $f_1 f_2 = O(g_1 g_2).$

Same Order

When $x \to c$, if $f$ and $g$ are of the same order, denoted $f = \Theta(g), \quad x \to c,$ it means:

$$\begin{equation*} f = O(g) \text{ and } g = O(f), \end{equation*}$$

that is, there exist $M > 0$ and a deleted neighborhood $W$ of $c$ such that for all $x \in W$,

$$\begin{equation*} \frac{1}{M} |g(x)| \le |f(x)| \le M |g(x)|. \end{equation*}$$

Small o

When $x \to c$, if $f(x) = o(g(x))$, it means:

For every $\varepsilon > 0$, there exists a deleted neighborhood $W$ of $c$ such that for all $x \in W$,

$$\begin{equation*} |f(x)| \le \varepsilon |g(x)|. \end{equation*}$$

• If $f_1 = o(g)$ and $f_2 = o(g)$ as $x \to c$, then $f_1 + f_2 = o(g).$

• If $f_1 = o(g_1)$ and $f_2 = O(g_2)$ as $x \to c$, then $f_1 f_2 = o(g_1 g_2).$

Asymptotic Equivalence

When $x \to c$, functions $f$ and $g$ are said to be asymptotically equivalent, written as $f \sim g$, if

$$\begin{equation*} f = g + o(g). \end{equation*}$$

That is, for every $\varepsilon > 0$, there exists a deleted neighborhood $W$ of $c$ such that for all $x \in W$,

$$\begin{equation*} |f(x) - g(x)| \le \varepsilon |g(x)|. \end{equation*}$$

Theorem 2.8. If $x \to c$, and $f + o(f) = G + o(g), \quad G = O(g),$ then $f = G + o(g)$.

Corollary 1. If $x \to c$, and $f$ is equivalent to $g$, then $g$ is equivalent to $f$.

Corollary 2. If $x \to c$, and $f$ is equivalent to $g$, and $g$ is equivalent to $h$, then $f$ is equivalent to $h$.

2.6 Continuous Functions

Continuous Function

Let $f: [a, b] \to \mathbf{R}$. We say that the function $f$ is continuous at the point $x_0 \in (a, b)$ if

$$\begin{equation*} \lim_{x \to x_0} f(x) = f(x_0). \end{equation*}$$

That is to say, for any given $\varepsilon > 0$, there exists a suitable $\delta > 0$, such that when $|x - x_0| < \delta$, we have

$$\begin{equation*} |f(x) - f(x_0)| < \varepsilon. \end{equation*}$$

• We say that $f$ is right-continuous at $x_0$ if $\forall \varepsilon > 0$, $\exists \delta > 0$ such that for all $x \in I$.

$$\begin{equation*} x_0 \le x < x_0 + \delta \;\Rightarrow\; |f(x) - f(x_0)| < \varepsilon. \end{equation*}$$

• Similarly, $f$ is left-continuous at $x_0$ if $\forall \varepsilon > 0$, $\exists \delta > 0$ such that for all $x \in I$.

$$\begin{equation*} x_0 - \delta < x \le x_0 \;\Rightarrow\; |f(x) - f(x_0)| < \varepsilon. \end{equation*}$$ $$\begin{equation*} f \text{ is continuous at } x_0 \;\Longleftrightarrow\; f \text{ is both right-continuous and left-continuous at } x_0. \end{equation*}$$

Theorem 2.9. If both $f$ and $g$ are continuous at $x_0$, then

$$\begin{equation*} f \pm g \text{ and } f g \text{ are continuous at } x_0. \end{equation*}$$

Moreover, if $g(x_0) \neq 0$, then

$$\begin{equation*} \dfrac{f}{g} \text{ is continuous at } x_0. \end{equation*}$$

Theorem 2.10. Let $f$ and $g$ be functions such that:

$$\begin{equation*} \begin{cases} f \text{ is continuous at } x_0, \\ g \text{ is continuous at } y_0 = f(x_0), \end{cases} \end{equation*}$$

then the composite function $g \circ f$ is continuous at $x_0$.

Theorem 2.11. Let $I, J \subseteq \mathbb{R}$ be intervals, and let $f: I \to J$ be monotone and onto. Then $f$ is a continuous function.

Removable Discontinuity

A point $x_0$ is called a removable discontinuity of a function $f$ if:

$\text{(1)}\quad x_0$ is a limit point of the domain $I$ of $f$, and the limit

$$\begin{equation*} \lim_{x \to x_0} f(x) \end{equation*}$$

exists;

$\text{(2)}\quad f$ is either undefined at $x_0$, or

$$\begin{equation*} f(x_0) \ne \lim_{x \to x_0} f(x). \end{equation*}$$

• In other words, $f$ is not continuous at $x_0$, but it can be made continuous by appropriately redefining the value $f(x_0)$.

Jump Discontinuity

A point $x_0$ is called a jump discontinuity if both one-sided limits exist but are not equal:

$$\begin{equation*} \lim_{x \to x_0^-} f(x) \quad \text{and} \quad \lim_{x \to x_0^+} f(x). \end{equation*}$$

Classification

• Type I discontinuities: include both removable and jump discontinuities.

• Type II discontinuities: all other discontinuities that do not fall into the above categories.

2.7 Uniform Continuity of Functions

Uniform Continuity

A function $f$ is said to be uniformly continuous on a set $K$, if for every $\varepsilon > 0$, there exists a $\delta_\varepsilon > 0$ such that

$$\begin{equation*} \forall x, y \in K, \quad |x - y| < \delta_\varepsilon \implies |f(x) - f(y)| < \varepsilon. \end{equation*}$$

• The function $f$ is not uniformly continuous on a set $K$ if and only if there exists an $\varepsilon_0 > 0$ such that for every $n \in \mathbb{N}^*$, one can find two points in $K$, denoted as $s_n$ and $t_n$, such that although $|s_n - t_n| < 1/n$, we have $$\begin{equation*} |f(s_n) - f(t_n)| \geqslant \varepsilon_0. \end{equation*}$$

2.8 Properties of Continuous Functions on Bounded Closed Intervals

Theorem 2.12. Let the function $f$ be continuous on $[a, b]$, then $f$ must be uniformly continuous on $[a, b]$.

Theorem 2.13. A continuous function on a bounded closed interval must be bounded on that interval.

Theorem 2.14. Let $f$ be continuous on $[a, b]$. Let

$$\begin{equation*} M = \sup_{x \in [a, b]} f(x), \quad m = \inf_{x \in [a, b]} f(x), \end{equation*}$$

then there must exist $x^*, x_* \in [a, b]$, such that

$$\begin{equation*} f(x^*) = M, \quad f(x_*) = m. \end{equation*}$$

Theorem 2.15. If $f$ is continuous on the interval $[a,b]$ and $f(a) \ne f(b)$, then every real number between $f(a)$ and $f(b)$ is attained as a function value of $f$ on $[a,b]$.