Continuity and Differentiability
Continuity
- Definition: A function is continuous on an interval if the graph of the function is unbroken, without jumps or missing points.
- Example: A function is continuous from (a) to (b) if there are no breaks or jumps.
- Types of Discontinuities:
- Jump Discontinuity: No connection between parts of the graph (non-removable).
- Hole (Removable Discontinuity): A missing point in the graph.
- Infinite Discontinuity: Occurs at a vertical asymptote (e.g., ( \frac{1}{x-2} ) with asymptote at ( x = 2 )).
Differentiability
- Definition: Describes whether the first derivative of a function is continuous.
- Relation to Continuity:
- If a function is continuous, it might not be differentiable.
- Differentiability implies the function is smooth, without sharp turns.
Examples of Differentiability
- Continuous and Differentiable Function: Smooth curve with no sharp turns.
- Absolute Value Function (( |x| )): Continuous everywhere but not differentiable at ( x = 0 ) due to a sharp turn.
Piecewise Functions
- To check continuity at a point, verify if left-side and right-side limits are equal.
- Differentiability requires that the derivative is continuous at the point.
- Example:
- Function: ( f(x) = x^2 \text{ for } x < 0; ; x+2 \text{ for } x \geq 0 )
- Continuity: Not continuous at ( x = 0 ) because limits from the left and right differ.
- Differentiability: Not differentiable at ( x = 0 ) due to lack of continuity.
Practice Problems
- For a piecewise function, check continuity and differentiability at given points by analyzing limits and derivative continuity.
- Graph piecewise functions to visualize continuity and differentiability.
Graphical Analysis
- Discontinuous Graphs: Show jumps, holes, or end asymptotes.
- Differentiable Functions: No sharp changes in direction, which maintains continuity in derivatives.
Key Concepts:
- Continuity Test: Analyze limits from both sides and ensure function values align.
- Differentiability Test: Ensure the first derivative is continuous.
- Vertical Tangents: Indicate potential non-differentiability due to infinite slopes.
Special Functions
- ( x^{1/3} ) and ( x^{2/3} )
- Both are continuous.
- Not differentiable at ( x = 0 ) due to vertical tangents.
- Derivatives indicate undefined conditions at ( x = 0 ).
Understanding Vertical Tangents
- Vertical Tangent: A slope of a vertical line (undefined), leading to non-differentiability.
- First Derivative: Evaluates continuity of the derivative, shows discontinuity like vertical asymptotes.
In summary, continuity is about unbroken graphs, while differentiability ensures smooth curves and consistent slopes. Special attention must be given to piecewise and special functions, especially at points of discontinuity or sharp turns.