Learning Outcomes

  1. Explain three conditions for continuity at a point 
  2. Describe three types of discontinuities
  3. Define continuity on an interval
  4. Understand the Intermediate Value Theorem

Textbook

  • Chapter 2.4  Continuity  

Textbook Assignment

  • p. 191:    131, 133, 139, 143, 145, 147

WeBWorK Assginment

  • Limits-Continuity

Exit problems of the session

  1. Is the function f(x) = \begin{cases}\dfrac{x+1}{x^2+3x+2}, & \text{ if } -5<x<-1 \\ 3x+5, & \text{ if } -1\leq x\leq 3 \end{cases}  continuous at x=-1 ? If not, what type of discontinuity is it?
  2. Find the value k that makes the following function continuous at the interval.   g(x) = \begin{cases}\sqrt{kx}, & \text{ if } 0\leq x \leq 3 \\ x+1, & \text{ if } 3 < x \leq 10 \end{cases} 

 

 Key Concepts

  • For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point.
  • Discontinuities may be classified as removable, jump, or infinite.
  • A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.
  • The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints.
 

Videos and Practice Problems of Selected Topics

  1. \rhd Types of discontinuity (7:15) A function being continuous at a point means that the two-sided limit at that point exists and is equal to the function’s value. Point/removable discontinuity is when the two-sided limit exists, but isn’t equal to the function’s value. Jump discontinuity is when the two-sided limit doesn’t exist because the one-sided limits aren’t equal. Asymptotic/infinite discontinuity is when the two-sided limit doesn’t exist because it’s unbounded.
  2. * Practice: Classify discontinuities. (4 problems)
  3. \rhd Continuity at a point (8:15) Saying a function f is continuous when x=a is the same as saying that the function’s two-side limit at x=a exists and is equal to f(a).
  4. \rhd Worked example: continuity at a point (graphical) (7:18). Two examples are given where the conditions for continuity at a point given a function’s graph are analyzed.
  5. * Practice: Continuity at a point (graphical). (4 problems)
  6. \rhd Worked example: point where a function is continuous (3:58) Is the function g(x) = \begin{cases}\log(3x), & \text{ if } 0<x<3 \\ (4-x)\log(9), & \text{ if } x\geq 3 \end{cases} continuous at x=3?
  7. \rhd Worked example: point where a function isn’t continuous (4:02) Is the function f(x) = \begin{cases}\ln(x), & \text{ if } 0<x\leq 2\\ x^2\ln(x), & \text{ if } x> 2 \end{cases} continuous at x=2?
  8. * Practice: Continuity at a point (algebraic). (4 problems)
  9. \rhd  Limits by direct substitution (2:06) Recognize that 6x^2+5x-1 is a continuous function to find \displaystyle\lim_{x\to -1}(6x^2+5x-1).
  10. * Practice: Limits by direct substitution. (4 problems)