Continuity of Functions
2.1 Cardinality of Sets
Let \(\mathbb{N}^*\) be the set of all positive integers, and \[ N_n = \{1, 2, \cdots, n\}. \]
\(\text{(1)}\quad\)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 \[ S = \bigcup_{n=1}^{\infty} E_n, \] then \(S\) is an at most countable set.
2.2 Elementary Functions
\[ f(x) = c, \quad c \in \mathbb{R}. \]
Domain: \(\mathbb{R}\).
Range: \(\{c \}\).
\[ f(x) = x. \]
Domain: \(\mathbb{R}\).
Range: \(\mathbb{R}\).
\[ f(x) = a^x, \quad a>0,\; a\neq1. \]
- Domain: \(\mathbb{R}\).
- Range: \((0, +\infty)\).
- Special case: \(f(x) = e^x\) is the natural exponential function.
\[ f(x) = \log_a x, \quad a>0,\; a\neq1. \]
- Domain: \((0, +\infty)\).
- Range: \(\mathbb{R}\).
- Special case: \(f(x) = \ln x = \log_e x\).
- Inverse: \((\log_a x)^{-1} = a^x.\)
\[ f(x) = x^a = e ^ {a \ln x}, \quad a \in \mathbb{R}. \]
- Domain: \((0, +\infty)\).
- Range: \((0, +\infty)\).
- Typical examples: \(x^2,\; x^3,\; \sqrt{x},\; \dfrac{1}{x}\).
\[ \sin x,\; \cos x,\; \tan x = \frac{\sin x}{\cos x},\; \cot x = \frac{\cos x}{\sin x},\; \sec x = \frac{1}{\cos x},\; \csc x = \frac{1}{\sin x}. \]
- Domain and Range:
| Function | Domain | Range |
|---|---|---|
| \(\sin x\) | \(\mathbb{R}\) | \([-1,1]\) |
| \(\cos x\) | \(\mathbb{R}\) | \([-1,1]\) |
| \(\tan x\) | \(x\neq \pi/2 + k\pi\) | \(\mathbb{R}\) |
\[ \sinh x = \frac{e^x - e^{-x}}{2}, \qquad \cosh x = \frac{e^x + e^{-x}}{2}, \qquad \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}. \]
- 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:
\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\\\).
\(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\\\).
\(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}.\)
\(\sin(2A)=2\sin A\cos A\).
\(\cos(2A)=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A\).
\(\tan(2A)=\frac{2\tan A}{1-\tan^2A}\).
\(\sin A+\sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}\).
\(\sin A-\sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}\).
\(\cos A+\cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}\).
\(\cos A-\cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}\).
\(\sin A\sin B = \tfrac{1}{2}\big[\cos(A-B)-\cos(A+B)\big]\).
\(\cos A\cos B = \tfrac{1}{2}\big[\cos(A-B)+\cos(A+B)\big]\).
\(\sin A\cos B = \tfrac{1}{2}\big[\sin(A+B)+\sin(A-B)\big]\).
For angles \(\theta\) in the interval \((0,\tfrac{\pi}{2}):\) \(\sin\theta < \theta < \tan\theta\).
\(\sin(2x) = 2 \sin x \cos x\).
\(\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x\).
\(\sinh(2x) = 2 \sinh x \cosh x\).
\(\cosh(2x) = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 1 + 2\sinh^2 x\).
\(\operatorname{asinh}(x) = \ln\!\left(x + \sqrt{x^2 + 1}\right)\).
\(\operatorname{acosh}(x) = \ln\!\left(x + \sqrt{x - 1}\sqrt{x + 1}\right)\).
\[ \arcsin x,\quad \arccos x,\quad \arctan x. \]
- Domain and Range:
| Function | Domain | Range |
|---|---|---|
| \(\arcsin x\) | \([-1,1]\) | \([-\pi/2,\ \pi/2]\) |
| \(\arccos x\) | \([-1,1]\) | \([0,\ \pi]\) |
| \(\arctan x\) | \(\mathbb{R}\) | \((-\pi/2,\ \pi/2)\) |
If \(f,g\) are elementary functions, then \[ f+g,\quad f-g,\quad f\cdot g,\quad \frac{f}{g}\;(g\neq0),\quad f\circ g \] are all elementary functions (in their domains of definition).
2.3 Limit of Functions
A real number \(x_0\) is called an accumulation point (or limit point) of a set \(I \subseteq \mathbb{R}\) if:
\[ \forall \varepsilon > 0,\; \exists x \in I \text{ such that } 0 < |x - x_0| < \varepsilon. \]
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 \[ \lim_{x \to x_0} f(x) = A. \]
if and only if
\[ \forall \varepsilon > 0,\; \exists \delta > 0 \text{ such that } \forall x \in I,\; 0 < |x - x_0| < \delta \Rightarrow |f(x) - A| < \varepsilon. \]
The right-hand limit of a function \(f\) at \(x_0\) is denoted by \[ \lim_{x \to x_0^+} f(x) = A, \] which means: \[ \forall \varepsilon > 0,\; \exists \delta > 0 \text{ such that } \forall x \in I,\; x_0 < x < x_0 + \delta \Rightarrow |f(x) - A| < \varepsilon. \]
Similarly, the left-hand limit is defined by \[ \lim_{x \to x_0^-} f(x) = A. \]
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 \(\lim\limits_{x \to x_0} g(x) \neq 0\).
\[ B_\delta(x_0) = \{x : |x - x_0| < \delta\}, \quad B_\delta(\mathring{x}_0) = \{x : 0 < |x - x_0| < \delta\}. \]
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 \[ \lim_{t \to t_0} f(g(t)) = l. \]
2.4 Infinite Limits of Functions
\[ \lim_{x \to +\infty} f(x) = A \]
means that
\[ \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; x > M \Rightarrow |f(x) - A| < \varepsilon. \]
Similarly, \[ \lim_{x \to -\infty} f(x) = A \]
means
\[ \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; x < -M \Rightarrow |f(x) - A| < \varepsilon. \]
\[ \lim_{x \to \infty} f(x) = A \]
means
\[ \forall \varepsilon > 0,\; \exists M > 0 \text{ such that } \forall x \in I,\; |x| > M \Rightarrow |f(x) - A| < \varepsilon. \]
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 \(x_0\) | \(J \setminus \{x_0\}\) |
| \(x_0^+\) | \([x_0,\, x_0+\delta)\) | \((x_0,\, x_0+\delta)\) |
| \(x_0^-\) | \((x_0-\delta,\, x_0]\) | \((x_0-\delta,\, x_0)\) |
| \(-\infty\) | \((-\infty,\, b)\) | \((-\infty,\, b)\) |
| \(+\infty\) | \((a,\, +\infty)\) | \((a,\, +\infty)\) |
| \(\infty\) | \((-\infty,\, b)\,\cup\,(a,\, +\infty)\) | \((-\infty,\, b)\,\cup\,(a,\, +\infty)\) |
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 \(f(x) \in V\).
When \(x \to c\), if \(f(x)\) grows without bound, we say \(f(x)\) is an infinite quantity (or positive/negative infinite quantity).
If \[ \lim_{x \to c} f(x) = \infty \quad \text{(i.e. } \lim_{x \to c} f(x) = +\infty \text{ or } \lim_{x \to c} f(x) = -\infty \text{)}, \] then \(f(x)\) diverges to infinity as \(x\) approaches \(c\).
Theorem 2.7 \[ \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e. \]
2.5 Big O and Small 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 \(f_1 f_2 = O(g_1 g_2).\)
When \(x \to c\), if \(f\) and \(g\) are of the same order, denoted \(f = \Theta(g), \quad x \to c,\) it means:
\[ f = O(g) \text{ and } g = O(f), \]
that is, there exist \(M > 0\) and a deleted neighborhood \(W\) of \(c\) such that for all \(x \in W\), \[ \frac{1}{M} |g(x)| \le |f(x)| \le M |g(x)|. \]
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 \(x \in W\), \[ |f(x)| \le \varepsilon |g(x)|. \]
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 \(f_1 f_2 = o(g_1 g_2).\)
When \(x \to c\), functions \(f\) and \(g\) are said to be asymptotically equivalent, written as \(f \sim g\), if \[ f = g + o(g). \]
That is, for every \(\varepsilon > 0\), there exists a deleted neighborhood \(W\) of \(c\) such that for all \(x \in W\), \[ |f(x) - g(x)| \le \varepsilon |g(x)|. \]
Theorem 2.8 If \(x \to c\), and \(f + o(f) = G + o(g), \quad G = O(g),\) then \(f = G + o(g)\).
Corollary 2.1 If \(x \to c\), and \(f\) is equivalent to \(g\), then \(g\) is equivalent to \(f\).
Corollary 2.2 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
Let \(f: [a, b] \to \mathbf{R}\). We say that the function \(f\) is continuous at the point \(x_0 \in (a, b)\) if \[ \lim_{x \to x_0} f(x) = f(x_0). \]
That is to say, for any given \(\varepsilon > 0\), there exists a suitable \(\delta > 0\), such that when \(|x - x_0| < \delta\), we have \[ |f(x) - f(x_0)| < \varepsilon. \]
- We say that \(f\) is right-continuous at \(x_0\) if \(\forall \varepsilon > 0\), \(\exists \delta > 0\) such that for all \(x \in I\).
\[ x_0 \le x < x_0 + \delta \;\Rightarrow\; |f(x) - f(x_0)| < \varepsilon. \]
- Similarly, \(f\) is left-continuous at \(x_0\) if \(\forall \varepsilon > 0\), \(\exists \delta > 0\) such that for all \(x \in I\).
\[ x_0 - \delta < x \le x_0 \;\Rightarrow\; |f(x) - f(x_0)| < \varepsilon. \]
\[ f \text{ is continuous at } x_0 \;\Longleftrightarrow\; f \text{ is both right-continuous and left-continuous at } x_0. \]
Theorem 2.9 If both \(f\) and \(g\) are continuous at \(x_0\), then \[ f \pm g \text{ and } f g \text{ are continuous at } x_0. \]
Moreover, if \(g(x_0) \neq 0\), then
\[ \dfrac{f}{g} \text{ is continuous at } x_0. \]
Theorem 2.10 Let \(f\) and \(g\) be functions such that: \[ \begin{cases} f \text{ is continuous at } x_0, \\ g \text{ is continuous at } y_0 = f(x_0), \end{cases} \] 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.
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 \[ \lim_{x \to x_0} f(x) \] exists;
\(\text{(2)}\quad f\) is either undefined at \(x_0\), or \[ f(x_0) \ne \lim_{x \to x_0} f(x). \]
- In other words, \(f\) is not continuous at \(x_0\), but it can be made continuous by appropriately redefining the value \(f(x_0)\).
A point \(x_0\) is called a jump discontinuity if both one-sided limits exist but are not equal: \[ \lim_{x \to x_0^-} f(x) \quad \text{and} \quad \lim_{x \to x_0^+} f(x). \]
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
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
\[ \forall x, y \in K, \quad |x - y| < \delta_\varepsilon \implies |f(x) - f(y)| < \varepsilon. \]
- 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 \[ |f(s_n) - f(t_n)| \geqslant \varepsilon_0. \]
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 \[ M = \sup_{x \in [a, b]} f(x), \quad m = \inf_{x \in [a, b]} f(x), \] then there must exist \(x^*, x_* \in [a, b]\), such that \[ f(x^*) = M, \quad f(x_*) = m. \]
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 \([a,b]\).