Learning Outcomes

- Explain three conditions for continuity at a point
- Describe three types of discontinuities
- Define continuity on an interval
- 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 **

- 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?
- 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
at that point, its**defined**at the point, and the value of the function at that point must*limit must exist*the value of the limit at that point.**equal** - Discontinuities may be classified as
,*removable*, or*jump*.*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

- $\rhd$ 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.
- * Practice: Classify discontinuities. (4 problems)
- $\rhd$ 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 $f(a)$.
- $\rhd$ 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.
- * Practice: Continuity at a point (graphical). (4 problems)
- $\rhd$ 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 $x=3$?
- $\rhd$ 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 $x=2$?
- * Practice: Continuity at a point (algebraic). (4 problems)
- $\rhd$ Limits by direct substitution (2:06) Recognize that $6x^2+5x-1$ is a continuous function to find $\displaystyle\lim_{x\to -1}(6x^2+5x-1)$.
- * Practice: Limits by direct substitution. (4 problems)