# Calculus

## Limits and Continuity of Functions # Continuity of Functions

### Heine Definition of Continuity

A real function $$f\left( x \right)$$ is said to be continuous at $$a \in \mathbb{R}$$ ($$\mathbb{R}-$$ is the set of real numbers), if for any sequence $$\left\{ {{x_n}} \right\}$$ such that

$\lim\limits_{n \to \infty } {x_n} = a,$

it holds that

$\lim\limits_{n \to \infty } f\left( {{x_n}} \right) = f\left( a \right).$

In practice, it is convenient to use the following three conditions of continuity of a function $$f\left( x \right)$$ at point $$x = a:$$

• Function $$f\left( x \right)$$ is defined at $$x = a;$$
• Limit $$\lim\limits_{x \to a} f\left( x \right)$$ exists;
• It holds that $$\lim\limits_{x \to a} f\left( x \right) = f\left( a \right).$$

### Cauchy Definition of Continuity $$\left(\varepsilon – \delta -\right.$$ Definition)

Consider a function $$f\left( x \right)$$ that maps a set $$\mathbb{R}$$ of real numbers to another set $$B$$ of real numbers. The function $$f\left( x \right)$$ is said to be continuous at $$a \in \mathbb{R}$$ if for any number $$\varepsilon \gt 0$$ there exists some number $$\delta \gt 0$$ such that for all $$x \in \mathbb{R}$$ with

$\left| {x – a} \right| \lt \delta ,$

the value of $$f\left( x \right)$$ satisfies:

$\left| {f\left( x \right) – f\left( a \right)} \right| \lt \varepsilon .$

### Definition of Continuity in Terms of Differences of Independent Variable and Function

We can also define continuity using differences of independent variable and function. The function $$f\left( x \right)$$ is said to be continuous at the point $$x = a$$ if the following is valid:

${\lim\limits_{\Delta x \to 0} \Delta y }={ \lim\limits_{\Delta x \to 0} \left[ {f\left( {a + \Delta x} \right) – f\left( a \right)} \right] }={ 0,}$

where $$\Delta x = x – a.$$

All the definitions of continuity given above are equivalent on the set of real numbers.

A function $$f\left( x \right)$$ is continuous on a given interval, if it is continuous at every point of the interval.

### Continuity Theorems

#### Theorem $$1.$$

Let the function $$f\left( x \right)$$ be continuous at $$x = a$$ and let $$C$$ be a constant. Then the function $$Cf\left( x \right)$$ is also continuous at $$x = a$$.

#### Theorem $$2.$$

Let the functions $${f\left( x \right)}$$ and $${g\left( x \right)}$$ be continuous at $$x = a$$. Then the sum of the functions $${f\left( x \right)} + {g\left( x \right)}$$ is also continuous at $$x = a.$$

#### Theorem $$3.$$

Let the functions $${f\left( x \right)}$$ and $${g\left( x \right)}$$ be continuous at $$x = a.$$ Then the product of the functions $${f\left( x \right)}{g\left( x \right)}$$ is also continuous at $$x = a.$$

#### Theorem $$4.$$

Let the functions $${f\left( x \right)}$$ and $${g\left( x \right)}$$ be continuous at $$x = a$$. Then the quotient of the functions $$\large\frac{{f\left( x \right)}}{{g\left( x \right)}} \normalsize$$ is also continuous at $$x = a$$ assuming that $${g\left( a \right)} \ne 0$$.

#### Theorem $$5.$$

Let $${f\left( x \right)}$$ be differentiable at the point $$x = a.$$ Then the function $${f\left( x \right)}$$ is continuous at that point.

Remark: The converse of the theorem is not true, that is, a function that is continuous at a point is not necessarily differentiable at that point.

#### Theorem $$6$$ (Extreme Value Theorem).

If $${f\left( x \right)}$$ is continuous on the closed, bounded interval $$\left[ {a,b} \right]$$, then it is bounded above and below in that interval. That is, there exist numbers $$m$$ and $$M$$ such that

$m \le f\left( x \right) \le M$

for every $$x$$ in $$\left[ {a,b} \right]$$ (see Figure $$1$$).

#### Theorem $$7$$ (Intermediate Value Theorem).

Let $${f\left( x \right)}$$ be continuous on the closed, bounded interval $$\left[ {a,b} \right]$$. Then if $$c$$ is any number between $${f\left( a \right)}$$ and $${f\left( b \right)}$$, there is a number $${x_0}$$ such that

$f\left( {{x_0}} \right) = c.$

The intermediate value theorem is illustrated in Figure $$2.$$

### Continuity of Elementary Functions

All elementary functions are continuous at any point where they are defined.

An elementary function is a function built from a finite number of compositions and combinations using the four operations (addition, subtraction, multiplication, and division) over basic elementary functions. The set of basic elementary functions includes:

1. Algebraical polynomials $$A{x^n} + B{x^{n – 1}} + \ldots$$ $$+ Kx + L;$$
2. Rational fractions $$\large\frac{{A{x^n} + B{x^{n – 1}} + \ldots + Kx + L}}{{M{x^m} + N{x^{m – 1}} + \ldots + Tx + U}}\normalsize$$;
3. Power functions $${x^p}$$;
4. Exponential functions $${a^x}$$;
5. Logarithmic functions $${\log _a}x$$;
6. Trigonometric functions $$\sin x$$, $$\cos x$$, $$\tan x$$, $$\cot x$$, $$\sec x$$, $$\csc x$$;
7. Inverse trigonometric functions $$\arcsin x$$, $$\arccos x$$, $$\arctan x$$, $$\text{arccot }x$$, $$\text{arcsec }x$$, $$\text{arccsc }x$$;
8. Hyperbolic functions $$\sinh x$$, $$\cosh x$$, $$\tanh x$$, $$\coth x$$, $$\text{sech }x$$, $$\text{csch }x$$;
9. Inverse hyperbolic functions $$\text{arcsinh }x$$, $$\text{arccosh }x$$, $$\text{arctanh }x$$, $$\text{arccoth }x,$$ $$\text{arcsech }x$$, $$\text{arccsch }x$$.

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Using the Heine definition, prove that the function $$f\left( x \right) = {x^2}$$ is continuous at any point $$x = a.$$

### Example 2

Using the Heine definition, show that the function $$f\left( x \right) = \sec x$$ is continuous for any $$x$$ in its domain.

### Example 3

Using Cauchy definition, prove that $$\lim\limits_{x \to 4} \sqrt x = 2$$.

### Example 4

Show that the cubic equation $$2{x^3} – 3{x^2} – 15 = 0$$ has a solution in the interval $$\left( {2,3} \right)$$.

### Example 5

Show that the equation $${x^{1000}} + 1000x – 1 = 0$$ has a root.

### Example 6

Let
${f\left(x \right) \text{=}}\kern0pt {\begin{cases} x^2 + 2, & x \lt 0 \\ ax + b, & 0 \le x \lt 1 \\ 3 + 2x – {x^2}, & x \ge 1 \end{cases}}$
Determine $$a$$ and $$b$$ so that the function $$f\left(x \right)$$ is continuous everywhere.

### Example 7

If the function
${f\left(x \right) \text{ = }}\kern0pt {\begin{cases} \cos \left( {2\pi x- a} \right), &x \lt -1 \\ x^3 + 1, &x \ge -1 \end{cases}}$
is continuous, what is the value of $$a?$$

### Example 1.

Using the Heine definition, prove that the function $$f\left( x \right) = {x^2}$$ is continuous at any point $$x = a.$$

Solution.

Using the Heine definition we can write the condition of continuity as follows:

${\lim\limits_{\Delta x \to 0} f\left( {a + \Delta x} \right) = f\left( a \right)\;\;\;}\kern-0.3pt {\text{or}\;\;\lim\limits_{\Delta x \to 0} \left[ {f\left( {a + \Delta x} \right) – f\left( a \right)} \right] } = {\lim\limits_{\Delta x \to 0} \Delta y = 0,}$

where $$\Delta x$$ and $$\Delta y$$ are small numbers shown in Figure $$3.$$

At any point $$x = a:$$
${f\left( a \right) = {a^2},\;\;\;}\kern-0.3pt{f\left( {a + \Delta x} \right) = {\left( {a + \Delta x} \right)^2}.}$
$\require{cancel} {\Delta y = f\left( {a + \Delta x} \right) – f\left( a \right) } = {{\left( {a + \Delta x} \right)^2} – {a^2} } = {\cancel{a^2} + 2a\Delta x + {\left( {\Delta x} \right)^2} – \cancel{a^2} } = {2a\Delta x + {\left( {\Delta x} \right)^2}.}$