Continuity

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 $f$ is continuous at a number a, if: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

• 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:
$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

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.

• Introduction
  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 $f$ is continuous at a number a, if: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$
  
  Classify different types of discontinuities: removable VS. infinite VS. jump
  
  Analyze: rational function with a hole: $f(x)=\frac{(x-1)(x-2)}{(x-2)}$
  VS. rational function with an asymptote: $g(x)=\frac{1}{x+1}$
  VS. piecewise function:
  
  
  c)

  Lesson Overview - Continuity
  
  

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• 1.
  Discussing “Continuity” Graphically
  The graph of a function $f$ is shown below.
  a)

  State the numbers at which $f$ is discontinuous.
  
  
  b)

  Explain and classify each discontinuity.
  
  
  
  

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

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• 3.
  Discontinuities of Rational Functions (denominator=0)
  Locate and classify each discontinuity of the function: $f(x)=\frac{x^2-9}{x^2+x-6}$

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• 4.
  Discussing “Continuity” Algebraically
  Find the values of a and b that make the function $f$ continuous on ($-\infty$, $\infty$).
  
  

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