AP Calculus BC / Unit 1: Limits and Continuity
1.12 Confirming Continuity over an Interval
Notes
A function is continuous on an interval if it's continuous at every point in it. Polynomials, exponentials, $\sin$ and $\cos$ are continuous everywhere. Rational, log and root functions are continuous on their domains.
- On a closed interval $[a,b]$, check one-sided continuity at the endpoints.
Practice
On which interval is $\displaystyle f(x)=\sqrt{x-2}$ continuous?
It's defined and continuous for $x\ge2$, with right-continuity at 2.
Where is $\displaystyle g(x)=\frac{x+1}{x^2-9}$ continuous?
Rational functions are continuous where the denominator isn't 0.
Which function is continuous for all real numbers?
Sums of everywhere-continuous functions are continuous everywhere.
Ready for more? Take the Unit 1 test.