1.8.1: Resources and Key Concepts

Resources

Videos

• Continuity
  • Limits - Continuity of Functions - Definition, Theory, and Examples
  • Limits - Continuity of Functions - Continuity on Intervals - Proving Arithmetic Retains Continuity
• Discontinuity
  • Limits - Continuity of Functions - Types of Discontinuities
• Continuity on an interval
  • Limits - Continuity of Functions - Continuity on Intervals
• Composite Function Theorem
  • Limits - Continuity of Functions - Continuity of Compositions - Proof: Limit(Comp) = Comp(Limit)
• Continuity of composite functions
  • Limits - Continuity of Functions - Continuity of Compositions
  • Limits - Continuity of Functions - Continuity of Compositions - Proof: Continuity of Compositions
• Intermediate Value Theorem
  • Limits - Continuity of Functions - The Intermediate Value Theorem
• Piecewise function continuity
  • Limits - Continuity of Functions - Example: Proving Continuity at 0 Without Using Squeeze Theorem

Key Concepts

Definitions

• Continuous at a Point: A function \(f(x)\) is said to be continuous at a point \(a\) if each of the following three conditions is satisfied:
  1. \(f(a)\) is defined
  2. \( \displaystyle \lim_{x \to a} f(x)\) exists
  3. \( \displaystyle \lim_{x \to a} f(x) = f(a)\)
• Discontinuous at a Point: A function is discontinuous at a point \(a\) if it fails to be continuous at \(a\).
• Types of Discontinuities: If \(f(x)\) is discontinuous at \(a\), then
  1. \(f\) has a removable discontinuity at \(a\) if \( \displaystyle \lim_{x \to a} f(x)\) exists. (Note: When we state that \( \displaystyle \lim_{x \to a} f(x)\) exists, we mean that \( \displaystyle \lim_{x \to a} f(x) = L\), where \(L\) is a real number.)
  2. \(f\) has a jump discontinuity at \(a\) if \( \displaystyle \lim_{x \to a^-} f(x)\) and \( \displaystyle \lim_{x \to a^+} f(x)\) both exist, but \( \displaystyle \lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)\). (Note: When we state that \( \displaystyle \lim_{x \to a^-} f(x)\) and \( \displaystyle \lim_{x \to a^+} f(x)\) both exist, we mean that both are real-valued and that neither takes on the values \(\pm\infty\).)
  3. \(f\) has an infinite discontinuity at \(a\) if \( \displaystyle \lim_{x \to a^-} f(x) = \pm\infty\) or \( \displaystyle \lim_{x \to a^+} f(x) = \pm\infty\).
• Continuity from the Right: A function \(f(x)\) is said to be continuous from the right at \(a\) if \( \displaystyle \lim_{x \to a^+} f(x) = f(a)\).
• Continuity from the Left: A function \(f(x)\) is said to be continuous from the left at \(a\) if \( \displaystyle \lim_{x \to a^-} f(x) = f(a)\).
• Continuity over Intervals:
  • A function \(f(x)\) is continuous over an open interval if it is continuous at every point in the interval.
  • A function \(f(x)\) is continuous over a closed interval of the form \([a,b]\) if it is continuous at every point in \((a,b)\) and is continuous from the right at \(a\) and is continuous from the left at \(b\).

Theorems

• Theorem: Continuity of Polynomials and Rational Functions: Polynomials and rational functions are continuous at every point in their domains.
• Theorem: Composite Function Theorem: If \(f(x)\) is continuous at \(L\) and \( \displaystyle \lim_{x \to a} g(x) = L\), then \( \displaystyle \lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) = f(L)\).
• Theorem: Continuity of Trigonometric Functions: Trigonometric functions are continuous over their entire domains.
• Theorem: Intermediate Value Theorem (IVT): Let \(f\) be continuous over a closed, bounded interval \([a,b]\). If \(z\) is any real number between \(f(a)\) and \(f(b)\), then there is a number \(c\) in \([a,b]\) satisfying \(f(c)=z\).

Common Mistakes

• Intuition Flaws for Continuity: The intuitive notion of a function being continuous if its graph can be traced without lifting a pencil has flaws.
• Misapplying the Intermediate Value Theorem:
  • The IVT only allows us to conclude that we can find a value between \(f(a)\) and \(f(b)\); it doesn’t allow us to conclude that we can’t find other values (e.g., if \(f(a) > 0\) and \(f(b) > 0\), it doesn't mean there are no zeros in between).
  • The IVT does not apply if the function is not continuous over the interval.