# Calculus

## Limits and Continuity of Functions # Continuity of Functions

## Heine Definition of Continuity

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

$\left| {x - a} \right| \lt \delta ,$

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

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

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
$\frac{{A{x^n} + B{x^{n - 1}} + \ldots + Kx + L}}{{M{x^m} + N{x^{m - 1}} + \ldots + Tx + U}};$
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 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:

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

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

At any point $$x = a:$$

$f\left( a \right) = {a^2},\;\;f\left( {a + \Delta x} \right) = {\left( {a + \Delta x} \right)^2}.$

So that

$\Delta y = f\left( {a + \Delta x} \right) - f\left( a \right) = \left( {a + \Delta x} \right)^2 - {a^2} = \cancel{a^2} + 2a\Delta x + {\left( {\Delta x} \right)^2} - \cancel{a^2} = 2a\Delta x + {\left( {\Delta x} \right)^2}.$

### 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

$x = \frac{\pi }{2} + k\pi ,\;\;k = 0, \pm 1, \pm 2, \ldots ,$

where cosine is zero.

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

$\Delta y = \sec \left( {x + \Delta x} \right) - \sec x = \frac{1}{{\cos \left( {x + \Delta x} \right)}} - \frac{1}{{\cos x}} = \frac{{\cos - \cos \left( {x + \Delta x} \right)}}{{\cos \left( {x + \Delta x} \right)\cos x}} = \frac{{ - 2\sin \left( {x + \frac{{\Delta x}}{2}} \right)\sin \left( { - \frac{{\Delta x}}{2}} \right)}}{{\cos \left( {x + \Delta x} \right)\cos x}} = \frac{{2\sin \left( {x + \frac{{\Delta x}}{2}} \right)\sin \frac{{\Delta x}}{2}}}{{\cos \left( {x + \Delta x} \right)\cos x}}.$

Calculate the limit as $$\Delta x \to 0.$$

$\lim\limits_{\Delta x \to 0} \Delta y = \lim\limits_{\Delta x \to 0} \frac{{2\sin \left( {x + \frac{{\Delta x}}{2}} \right)\sin \frac{{\Delta x}}{2}}}{{\cos \left( {x + \Delta x} \right)\cos x}} = \lim\limits_{\Delta x \to 0} \frac{{2\sin \left( {x + \frac{{\Delta x}}{2}} \right)}}{{\cos \left( {x + \Delta x} \right)\cos x}} \cdot \lim\limits_{\Delta x \to 0} \sin \frac{{\Delta x}}{2} = \frac{{2\sin x}}{{{{\cos }^2}x}} \cdot 0 = 0.$

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

$x = \frac{\pi }{2} + k\pi ,\;\;k = 0, \pm 1, \pm 2, \ldots$

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

See more problems on Page 2.