Calculus

Limits and Continuity of Functions

Limits and Continuity Logo

Discontinuous Functions

  • If \(f\left( x \right)\) is not continuous at \(x = a\), then \(f\left( x \right)\) is said to be discontinuous at this point. Figures \(1 – 4\) show the graphs of four functions, two of which are continuous at \(x =a\) and two are not.

    example 1 of continuous function
    Fig 1. Continuous function.
    example 1 of discontinuous function
    Fig 2. Discontinuous function.
    example 2 of continuous function
    Fig 3. Continuous function.
    example 2 of discontinuous function
    Fig 4. Discontinuous function.

    Classification of Discontinuity Points

    All discontinuity points are divided into discontinuities of the first and second kind.

    The function \(f\left( x \right)\) has a discontinuity of the first kind at \(x = a\) if

    • There exist left-hand limit \(\lim\limits_{x \to a – 0} f\left( x \right)\) and right-hand limit \(\lim\limits_{x \to a + 0} f\left( x \right)\);
    • These one-sided limits are finite.

    Further there may be the following two options:

    • The right-hand limit and the left-hand limit are equal to each other: \[{\lim\limits_{x \to a – 0} f\left( x \right) }={ \lim\limits_{x \to a + 0} f\left( x \right).}\] Such a point is called a removable discontinuity.
    • The right-hand limit and the left-hand limit are unequal:\[{\lim\limits_{x \to a – 0} f\left( x \right) }\ne{ \lim\limits_{x \to a + 0} f\left( x \right).}\] In this case the function \(f\left( x \right)\) has a jump discontinuity.

    The function \(f\left( x \right)\) is said to have a discontinuity of the second kind (or a nonremovable or essential discontinuity) at \(x = a\), if at least one of the one-sided limits either does not exist or is infinite.


  • Solved Problems

    Click a problem to see the solution.

    Example 1

    Investigate continuity of the function \(f\left( x \right) = {3^{\large\frac{x}{{1 – {x^2}}}\normalsize}}.\)

    Example 2

    Show that the function \(f\left( x \right) = {\large\frac{{\sin x}}{x}\normalsize}\) has a removable discontinuity at \(x = 0.\)

    Example 3

    Find the points of discontinuity of the function \(f\left( x \right) = \begin{cases} 1 – {x^2}, & x \lt 0 \\ x +2, &x \ge 0 \end{cases} \) if they exist.

    Example 4

    Find the points of discontinuity of the function \(f\left( x \right) = \arctan {\large\frac{1}{x}\normalsize}\) if they exist.

    Example 5

    Find the points of discontinuity of the function \(f\left( x \right) = {\large\frac{{\left| {2x + 5} \right|}}{{2x + 5}}\normalsize}\) if they exist.

    Example 1.

    Investigate continuity of the function \(f\left( x \right) = {3^{\large\frac{x}{{1 – {x^2}}}\normalsize}}.\)

    Solution.

    The given function is not defined at \(x = -1\) and \(x = 1\). Hence, this function has discontinuities at \(x = \pm 1\). To determine the type of the discontinuities, we find the one-sided limits:

    \[
    {\lim\limits_{x \to – 1 – 0} {3^{\large\frac{x}{{1 – {x^2}}}\normalsize}} = {3^{\large\frac{{ – 1}}{{ – 0}}\normalsize}} }={ {3^\infty } = \infty ,\;\;\;}\kern-0.3pt
    {\lim\limits_{x \to – 1 + 0} {3^{\large\frac{x}{{1 – {x^2}}}\normalsize}} = {3^{\large\frac{{ – 1}}{{ + 0}}\normalsize}} }={ {3^{ – \infty }} = \frac{1}{{{3^\infty }}} = 0.}
    \]

    Since the left-side limit at \(x = -1\) is infinity, we have an essential discontinuity at this point.

    \[
    {\lim\limits_{x \to 1 – 0} {3^{\large\frac{x}{{1 – {x^2}}}\normalsize}} = {3^{\large\frac{{ 1}}{{ +0}}\normalsize}} }={ {3^\infty } = \infty ,\;\;\;}\kern-0.3pt
    {\lim\limits_{x \to 1 + 0} {3^{\large\frac{x}{{1 – {x^2}}}\normalsize}} = {3^{\large\frac{{ 1}}{{ -0}}\normalsize}} }={ {3^{ – \infty }} = \frac{1}{{{3^\infty }}} = 0.}
    \]

    Similarly, the right-side limit at \(x = 1\) is infinity. Hence, here we also have an essential discontinuity.

    Page 1
    Problem 1
    Page 2
    Problems 2-5