Calculus

Limits and Continuity of Functions

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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 \(x = a:\)

    • 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 \(\left(\varepsilon – \delta -\right.\) 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 \(1\)).

    extreme-value-theorem
    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 \(2.\)

    intermediate-value-theorem
    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 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 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 \(3.\)

    function-x2
    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

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

    Page 1
    Problem 1
    Page 2
    Problems 2-7