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# Calculus

Limits and Continuity of Functions

# Continuity of Functions

Page 1
Problem 1
Page 2
Problems 2-7

### 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 $$x = a:$$

1. Function $$f\left( x \right)$$ is defined at $$x = a$$;
2. Limit $$\lim\limits_{x \to a} f\left( x \right)$$ exists;
3. 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 $$1$$).

Figure 1.

Figure 2.

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 on problem description to see 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 $$x = a$$:
${f\left( a \right) = {a^2},\;\;\;}\kern-0.3pt{f\left( {a + \Delta x} \right) = {\left( {a + \Delta x} \right)^2}.}$ So that
$\require{cancel} {\Delta y = f\left( {a + \Delta x} \right) – f\left( a \right) } = {{\left( {a + \Delta x} \right)^2} – {a^2} } = {\cancel{a^2} + 2a\Delta x + {\left( {\Delta x} \right)^2} – \cancel{a^2} } = {2a\Delta x + {\left( {\Delta x} \right)^2}.}$

Calculate the limit:
${\lim\limits_{\Delta x \to 0} \Delta y = \lim\limits_{\Delta x \to 0} \left( {2a\Delta x + {{\left( {\Delta x} \right)}^2}} \right) } = {\lim\limits_{\Delta x \to 0} \left( {2a\Delta x} \right) + \lim\limits_{\Delta x \to 0} {\left( {\Delta x} \right)^2} } = {2a\lim\limits_{\Delta x \to 0} \Delta x + \lim\limits_{\Delta x \to 0} \Delta x \cdot \lim\limits_{\Delta x \to 0} \Delta x } = {2a \cdot 0 + 0 \cdot 0 = 0.}$ Thus, the function is continuous at any point $$x = a$$.

Figure 3.

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Problem 1
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Problems 2-7