Continuity of Functions

Heine Definition of Continuity

A real function f (x) is said to be continuous at a ∈ ℝ (ℝ − is the set of real numbers), if for any sequence {x_n} such that

it holds that

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

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

Cauchy Definition of Continuity $(\epsilon \text{-}\delta -$ 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 $\epsilon >0$ there exists some number $\delta >0$ such that for all $x\in \mathbb{R}$ with

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

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:

where $\mathrm{\Delta }x=x-a.$

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 $$ be continuous at $$ and let $$ be a constant. Then the function $$ is also continuous at $$

Theorem 2.

Let the functions $$ and $$ be continuous at $$. Then the sum of the functions $$ is also continuous at $$

Theorem 3.

Let the functions $$ and $$ be continuous at $$ Then the product of the functions $$ is also continuous at $$

Theorem 4.

Let the functions $$ and $$ be continuous at $$. Then the quotient of the functions $$ is also continuous at $$ assuming that $$

Theorem 5.

Let $$ be differentiable at the point $$ Then the function $$ 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 $$ is continuous on the closed, bounded interval $$, then it is bounded above and below in that interval. That is, there exist numbers $$ and $$ such that

for every $$ in $$ (see Figure $$).

Theorem 7 (Intermediate Value Theorem).

Let $$ be continuous on the closed, bounded interval $$. Then if $$ is any number between $$ and $$, there is a number $$ such that

The intermediate value theorem is illustrated in Figure $$

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
   $$
2. Rational fractions
   $$
3. Power functions $$;
4. Exponential functions $$;
5. Logarithmic functions $$;
6. Trigonometric functions
   $$
7. Inverse trigonometric functions
   $$
8. Hyperbolic functions
   $$
9. Inverse hyperbolic functions
   $$

Solved Problems

Solution.

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

where $$ and $$ are small numbers shown in Figure $$

At any point $$

So that

Solution.

The secant function $$ has domain all real numbers $$ except those of the form

where cosine is zero.

Let $$ be a differential of independent variable $$ Find the corresponding differential of function $$

Calculate the limit as $$

This result is valid for for all $$ except the roots of the cosine function:

Hence, the range of continuity and the domain of the function $$ fully coincide.