Continuity and Limits

Continuity of Functions

Let $f: I \to \mathbb{R}$ be a function and let $x_0 \in I$.

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

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

We say that $f$ is a continuous function if $f$ is continuous on its entire domain.

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 $x \in I$.

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

Meaning of Continuity

For $x$ sufficiently close to $x_0$, the function value $f(x)$ is approximately equal to $f(x_0)$:

$$\begin{equation*} f(x) \approx f(x_0) \quad \text{when } x \text{ is close to } x_0. \end{equation*}$$

Hence, continuity means that small changes in $x$ lead to small changes in $f(x)$.

Examples

1. Constant functions are continuous.

2. The function $f(x) = x^2$ is continuous on $\mathbb{R}$.

3. The function $f(x) = \sqrt{x}$ is continuous on the interval $[0, +\infty)$.

4. The function $f(x) = \frac{1}{x}$ is continuous on $\mathbb{R} \setminus \{0\}$.

Theorems

1. 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 $x_0$.

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

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

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

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

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

​ Corollary: The exponential function, the logarithmic function, and the power function are all continuous functions.

Limits of Functions

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

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

Theorems

1. Let $x_0$ be an accumulation point of $I$. Then

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

   if and only if the function

   $$\begin{equation*} g(x) = \begin{cases} f(x), & x \in I \setminus \{x_0\} \\ A, & x = x_0 \end{cases} \end{equation*}$$

   is continuous at $x_0$.

2. If

$$\begin{equation*} \lim_{x \to x_0} f(x) = A, \quad \lim_{x \to x_0} g(x) = B, \end{equation*}$$

​ then

$$\begin{equation*} \begin{aligned} \lim_{x \to x_0} \big(f(x) + g(x)\big) &= A + B, \\ \lim_{x \to x_0} \big(f(x)g(x)\big) &= AB. \end{aligned} \end{equation*}$$

​ If $B \neq 0$, then

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

3. if

$$\begin{equation*} \lim_{x \to x_0} f(x) = y_0, \quad \lim_{y \to y_0} g(y) = B, \quad \text{and } g \textbf{ is continuous at } y_0, \end{equation*}$$

then

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

4. Let $f, g, h$ be functions defined on an interval $I$, and let $x_0$ be a limit point of $I$. Suppose that for all $x \in I$,

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

If

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

then

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

Removable and Other Types of Discontinuities

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

1. $x_0$ 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;

2. $f$ 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 $f(x_0)$.

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.