- $\rhd$ 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)
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