AP Calculus AB
Advanced Placement Calculus AB covering limits, derivatives, and integrals.
Advanced Topics
Continuity and Types of Discontinuities
Understanding Continuity
A function is continuous if you can draw its graph without lifting your pencil. This means there are no gaps, jumps, or holes in the graph.
Types of Discontinuities
- Removable Discontinuity: A hole in the graph—often caused by a factor that cancels.
- Jump Discontinuity: The graph jumps from one value to another.
- Infinite Discontinuity: The graph goes off to infinity (vertical asymptote).
Why Does This Matter?
Calculus relies on continuity for derivatives and integrals to exist. Discontinuities can cause problems in calculations.
Identifying Discontinuities
- Look for division by zero.
- Check for places where the function isn't defined.
- Analyze the left and right limits.
Real-World Impact
Discontinuities can represent sudden changes, like a switch turning on, or an object changing direction.
Key Formula
\[f(x) = \frac{x^2-1}{x-1}\]
Examples
The graph of \( f(x) = \frac{x^2-1}{x-1} \) has a hole at \( x = 1 \).
A step function modeling elevator floors has jump discontinuities.
In a Nutshell
Continuity ensures smooth behavior; discontinuities reveal gaps, jumps, or infinite breaks.