Summary of "Continuity and Differentiability"

Main Ideas and Concepts

Continuity:
- A function is continuous on an interval if there are no breaks, jumps, or missing points in the graph.
- Types of discontinuities:
  - Jump Discontinuity: A sudden change in function values; the graph does not connect.
  - Removable Discontinuity: A "hole" in the graph where a limit exists but the function is not defined.
  - Infinite Discontinuity: Occurs at vertical asymptotes where the function approaches infinity.
Differentiability:
- A function is differentiable at a point if its derivative exists at that point.
- Differentiability is related to the Continuity of the first derivative.
- A function can be continuous but not differentiable if it has sharp turns or corners.
Graphical Examples:
- The Absolute Value Function is continuous but not differentiable at the vertex (sharp turn).
- Piecewise Functions require checking Continuity and Differentiability at the boundaries.
Three-Step Continuity Test:
- To determine Continuity at a point:
  - Check if the left-hand limit equals the right-hand limit.
  - Ensure the function is defined at that point.
  - Confirm that the limit equals the function value at that point.
Practice Problems:
- The video encourages viewers to solve practice problems involving Piecewise Functions and analyze their Continuity and Differentiability.

For Continuity:
- Evaluate limits from the left and right at the point of interest.
- Check if the function is defined at that point.
- Ensure the limits match the function value.
For Differentiability:
- If the function is continuous, find the first derivative.
- Check the Continuity of the first derivative using the same three-step test.
- If the first derivative is not continuous, the original function is not differentiable.

Absolute Value Function: Continuous everywhere but not differentiable at zero due to a sharp turn.
Piecewise Functions: Evaluated for Continuity and Differentiability at the joining point.
Functions with Vertical Tangents: Continuous but not differentiable at points where the slope is undefined.