# 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

it holds that

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 \(\left(\varepsilon \text{-} \delta -\right.\) 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

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 \(\Delta x = x - a.\)

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 \(\frac{{f\left( x \right)}}{{g\left( x \right)}}\) 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

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

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:

- Algebraical polynomials
\[A{x^n} + B{x^{n - 1}} + \ldots + Kx + L;\]
- Rational fractions
\[\frac{{A{x^n} + B{x^{n - 1}} + \ldots + Kx + L}}{{M{x^m} + N{x^{m - 1}} + \ldots + Tx + U}};\]
- Power functions \({x^p}\);
- Exponential functions \({a^x}\);
- Logarithmic functions \({\log_a}x\);
- Trigonometric functions
\[\sin x,\; \cos x,\; \tan x,\; \cot x,\; \sec x,\; \csc x;\]
- Inverse trigonometric functions
\[\arcsin x,\; \arccos x,\; \arctan x,\; \text{arccot}\,x,\; \text{arcsec}\,x,\; \text{arccsc}\,x;\]
- Hyperbolic functions
\[\sinh x,\; \cosh x,\; \tanh x,\; \coth x,\; \text{sech}\,x,\; \text{csch}\,x;\]
- 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 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:

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

At any point \(x = a:\)

So that

### Example 2.

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

Solution.

The secant function \(f\left( x \right) = \sec x = {\frac{1}{{\cos x}}}\) has domain all real numbers \(x\) except those of the form

where cosine is zero.

Let \(\Delta x\) be a differential of independent variable \(x.\) Find the corresponding differential of function \(\Delta y.\)

Calculate the limit as \(\Delta x \to 0.\)

This result is valid for for all \(x\) except the roots of the cosine function:

Hence, the range of continuity and the domain of the function \(f\left( x \right) = \sec x\) fully coincide.