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Intros
Lessons
  1. Introduction to Continuity
  2. Discuss CONTINUITY in everyday language: a function whose graph has no break in it.
  3. 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)
  4. Lesson Overview - Continuity
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Examples
Lessons
  1. Discussing "Continuity" Graphically
    The graph of a function ff is shown below.
    1. State the numbers at which ff is discontinuous.
    2. Explain and classify each discontinuity.
      Graph of a function and continuity
  2. 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
    1. 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}
      1. Discussing "Continuity" Algebraically
        Find the values of a and b that make the function ff continuous on (-\infty, \infty).
        Continuity