Continuity of Functions

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

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:

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

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.