Calculus

Limits and Continuity of Functions

Continuity of Functions

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Problem 1
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Problems 2-7

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.

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 \(1\)).

Illustration of Extreme Value Theorem

Figure 1.

Illustration of Intermediate Value Theorem

Figure 2.

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 \(\large\frac{{A{x^n} + B{x^{n – 1}} + \ldots + Kx + L}}{{M{x^m} + N{x^{m – 1}} + \ldots + Tx + U}}\normalsize\);
  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 \(x = a\).

 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 3

Using Cauchy definition, prove that \(\lim\limits_{x \to 4} \sqrt x = 2\).

 Example 4

Show that the cubic equation \(2{x^3} – 3{x^2} – 15 = 0\) has a solution in the interval \(\left( {2,3} \right)\).

 Example 5

Show that the equation \({x^{1000}} + 1000x – 1 = 0\) has a root.

 Example 6

Let

\[
{f\left(x \right) \text{=}}\kern0pt
{\begin{cases}
x^2 + 2, & x \lt 0 \\
ax + b, & 0 \le x \lt 1 \\
3 + 2x – {x^2}, & x \ge 1
\end{cases}}\]

Determine \(a\) and \(b\) so that the function \(f\left(x \right)\) is continuous everywhere.

 Example 7

If the function

\[
{f\left(x \right) \text{ = }}\kern0pt
{\begin{cases}
\cos \left( {2\pi x- a} \right), &x \lt -1 \\
x^3 + 1, &x \ge -1
\end{cases}}
\]

is continuous, what is the value of \(a?\)

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

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

Calculate the limit:

\[
{\lim\limits_{\Delta x \to 0} \Delta y = \lim\limits_{\Delta x \to 0} \left( {2a\Delta x + {{\left( {\Delta x} \right)}^2}} \right) }
= {\lim\limits_{\Delta x \to 0} \left( {2a\Delta x} \right) + \lim\limits_{\Delta x \to 0} {\left( {\Delta x} \right)^2} }
= {2a\lim\limits_{\Delta x \to 0} \Delta x + \lim\limits_{\Delta x \to 0} \Delta x \cdot \lim\limits_{\Delta x \to 0} \Delta x }
= {2a \cdot 0 + 0 \cdot 0 = 0.}
\]

Thus, the function is continuous at any point \(x = a\).

Continuity of the function y=x^2

Figure 3.

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Problem 1
Page 2
Problems 2-7