Overview
This lecture explains different types of discontinuities in functions, how to identify them on graphs and equations, and methods to find constants that make piecewise functions continuous.
Types of Continuity and Discontinuity
- A function is continuous if its graph has no jumps, breaks, or holes.
- A hole in the graph is called a removable discontinuity and occurs when a factor cancels in rational expressions.
- Jump discontinuity occurs when the function "jumps" to a different value at a point, often due to absolute value or piecewise functions.
- Infinite discontinuity occurs at a vertical asymptote, where the function approaches infinity on one or both sides.
- Only holes are removable discontinuities; jumps and infinite discontinuities are non-removable.
Identifying Discontinuities
- For rational functions, set the denominator to zero to find vertical asymptotes (infinite discontinuities).
- If a factor in the denominator cancels with the numerator, the x-value is a removable discontinuity (hole).
- Example: In 1/(x²), x = 0 is an infinite discontinuity.
- In (3x+2)/[(x+2)(x-5)], x = -2 is a hole (removable), x = 5 is an asymptote (infinite discontinuity).
- For f(x) = |x|/x, x = 0 is a jump discontinuity.
Continuity in Piecewise Functions
- Polynomial functions (linear, quadratic, cubic) are continuous everywhere.
- Discontinuities in piecewise functions only occur at the transition points.
- Check continuity at transition points by plugging the x-value into both defining equations; if y-values differ, the function is discontinuous there.
Making Piecewise Functions Continuous
- Set the expressions at each transition point equal to each other and solve for the unknown constant(s).
- Example: For f(x) = Cx+3 (x<2), 3x+C (x≥2), set 2C+3 = 6+C to get C = 3.
- For f(x) = ax-2 (x<3), x²-5 (x≥3), set 3a-2 = 4 to get a = 2.
- For more complex piecewise functions, set equal at each transition, substitute, and solve the resulting system of equations.
Key Terms & Definitions
- Continuity — a function with no jumps, breaks, or holes in its graph.
- Removable Discontinuity — a hole in the graph; can be "fixed" by redefining the function at that point.
- Non-Removable Discontinuity — discontinuities that cannot be fixed; includes jumps and infinite discontinuities.
- Jump Discontinuity — the function value "jumps" at a point; the left and right limits are different.
- Infinite Discontinuity — the function goes to infinity at a point; usually at a vertical asymptote.
- Piecewise Function — a function composed of different expressions over different intervals.
Action Items / Next Steps
- Practice identifying and classifying discontinuities on various graphs and equations.
- Complete exercises on finding values of constants that ensure continuity in piecewise functions.
- Review definitions and examples of each type of discontinuity.