Overview
This lecture covers the concept of continuity in functions, different types of discontinuities in graphs, and methods to identify and address points of discontinuity, including solving for continuity in piecewise functions.
Types of Continuity and Discontinuity
- A graph is continuous if it has no jumps, breaks, or holes.
- A hole in the graph is a removable discontinuity.
- A jump discontinuity occurs when the left and right sides of the graph do not connect.
- An infinite discontinuity occurs near a vertical asymptote, where the function approaches infinity on one or both sides.
Identifying Points of Discontinuity
- To find infinite discontinuities in rational functions, set the denominator equal to zero to find vertical asymptotes.
- Example: For (1/x^2), the graph is undefined at (x=0), giving an infinite discontinuity.
- Example: For (5/(x+2)), (x=-2) is the point of infinite discontinuity (vertical asymptote).
- If a factor cancels from both numerator and denominator (e.g., ((x+2))), the discontinuity at that x-value is a hole (removable).
Piecewise and Absolute Value Functions
- Absolute value functions like (f(x) = |x|/x) have a jump discontinuity at (x=0), which is non-removable.
- Piecewise functions are potentially discontinuous at the points where the pieces meet.
- To check for continuity at a boundary, plug the value into both pieces and compare y-values.
Making Piecewise Functions Continuous
- Set the outputs of pieces equal at intersection points and solve for unknown coefficients to ensure continuity.
- Example: For (f(x) = cx+3) for (x<2) and (f(x) = 3x+c) for (x \geq 2), set (2c+3 = 6+c) to find (c=3).
- For more complex functions, repeat this process at each joining point to solve for multiple unknowns.
Key Terms & Definitions
- Continuous Function — a function with no jumps, breaks, or holes in its graph.
- Removable Discontinuity (Hole) — a single missing point in the graph, often due to a factor cancelling in a rational function.
- Jump Discontinuity — when the graph “jumps” from one value to another at a point.
- Infinite Discontinuity — occurs at a vertical asymptote where the function graph heads toward infinity.
- Vertical Asymptote — a vertical line (x = a) where the function grows without bound.
Action Items / Next Steps
- Practice identifying discontinuities in various types of functions (rational, absolute value, piecewise).
- Solve for constants to ensure piecewise functions are continuous at boundary points.
- Review the definitions and examples of types of discontinuity for exam preparation.