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
\[\lim\limits_{n \to \infty } {x_n} = a,\] 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.

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 \(x = a\).

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
\[\require{cancel}
{\Delta y = f\left( {a + \Delta x} \right) – f\left( a \right) }
= {{\left( {a + \Delta x} \right)^2} – {a^2} }
= {\cancel{a^2} + 2a\Delta x + {\left( {\Delta x} \right)^2} – \cancel{a^2} }
= {2a\Delta x + {\left( {\Delta x} \right)^2}.}
\]