Continuity

Continuity

The idea of continuity lies in many things we experience in our daily lives, for instance, the time it takes you to log into StudyPug and read this section. Limits and continuity are so related that we cannot only learn about one and ignore the other. We will learn about the relationship between these two concepts in this section.

Lessons

• Definition of “continuity” in everyday language
A function is continuous if it has no holes, asymptotes, or breaks. A continuous graph can be drawn without removing your pen from the paper.

• Definition of “continuity” in Calculus
A function ff is continuous at a number a, if: limxaf(x)=limxa+f(x)=f(a)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)

• Polynomials are always continuous everywhere. Rational functions are continuous wherever the functions are defined; in other words, avoiding holes and asymptotes, rational functions are continuous everywhere. A function f is continuous at a number a, if and only if:
limxaf(x)=limxa+f(x)=f(a)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)
graph of a continuous function (a)
In simple words, the graph of a continuous function has no break in it and can be drawn without lifting your pen from the paper.
  • 1.
    Introduction to Continuity
    a)
    Discuss CONTINUITY in everyday language: a function whose graph has no break in it.

    b)
    Discuss CONTINUITY in the context of Calculus:
    A function ff is continuous at a number a, if: limxaf(x)=limxa+f(x)=f(a)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)

    Classify different types of discontinuities: removable VS. infinite VS. jump

    Analyze: rational function with a hole: f(x)=(x1)(x2)(x2)f(x)=\frac{(x-1)(x-2)}{(x-2)}
    VS.
    rational function with an asymptote: g(x)=1x+1g(x)=\frac{1}{x+1}
    VS.
    piecewise function: piecewise function h(x)

    c)
    Lesson Overview - Continuity


  • 2.
    Discussing “Continuity” Graphically
    The graph of a function ff is shown below.
    a)
    State the numbers at which ff is discontinuous.

    b)
    Explain and classify each discontinuity.
    Graph of a function and continuity


  • 3.
    Detecting Discontinuities Are the following functions continuous at x=3x=3?
    i) f(x)=x22x3x3f(x)=\frac{x^2-2x-3}{x-3}

    ii) Detecting Discontinuities 2
    iii) Detecting Discontinuities 3

  • 4.
    Discontinuities of Rational Functions (denominator=0)
    Locate and classify each discontinuity of the function: f(x)=x29x2+x6f(x)=\frac{x^2-9}{x^2+x-6}

  • 5.
    Discussing “Continuity” Algebraically
    Find the values of a and b that make the function ff continuous on (-\infty, \infty).
    Continuity