AP Calculus AB / Unit 1: Limits and Continuity
1.11 Defining Continuity at a Point
Notes
$f$ is continuous at $x=c$ if all three hold:
- $f(c)$ is defined.
- $\displaystyle \lim_{x\to c}f(x)$ exists.
- $\displaystyle \lim_{x\to c}f(x)=f(c)$.
Practice
Which is NOT a requirement for continuity at $x=c$?
Differentiability isn't required for continuity.
$f(x)=\begin{cases}x+1 & x\ne2\\ 5 & x=2\end{cases}$ is continuous at $x=2$?
Condition three fails.
For what value of $k$ is $f(x)=\begin{cases}x^2 & x<3\\ k & x=3\\ 3x & x>3\end{cases}$ continuous at 3?
Both sides approach 9, so set $k=9$.
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