Determine if the Piecewise Function is Continuous by using the Definition of Continuity  Summary and Q&A
TL;DR
The function is not continuous at x equals three because the limit as x approaches three does not exist.
Key Insights
 π Continuity of a function at a specific point is determined by three conditions: f(c) is defined, the limit as x approaches c exists, and the limit as x approaches c of f(x) equals f(c).
 π If any of the three conditions for continuity are not satisfied, the function is not continuous at the given point.
 βΊοΈ In this case, the function f(x) = x + 2 for x less than or equal to 3, and f(x) = 2x  5 for x greater than 3 is not continuous at x equals 3 because the second condition fails.
 π The limit as x approaches 3 from the left is 5, while the limit as x approaches 3 from the right is 1, indicating that the overall limit does not exist.
Questions & Answers
Q: What are the three conditions for continuity of a function?
The three conditions for continuity are: 1) f(c) is defined, 2) the limit as x approaches c exists, and 3) the limit as x approaches c of f(x) equals f(c).
Q: Why is it important to check if f(c) is defined?
It is important to check if f(c) is defined because for a function to be continuous at x equals c, plugging in c into the function should yield a defined value.
Q: How are the limits as x approaches 3 from the left and right calculated?
The limit as x approaches 3 from the left is calculated by replacing f(x) with x + 2 and evaluating the expression when x approaches 3. The limit as x approaches 3 from the right is calculated by replacing f(x) with 2x  5 and evaluating the expression when x approaches 3.
Q: Why does the function fail the second condition for continuity?
The function fails the second condition for continuity because the limits as x approaches 3 from the left and right are not equal. The left limit is 5 and the right limit is 1, indicating that the overall limit as x approaches 3 does not exist.
Summary & Key Takeaways

The function f(x) is checked for continuity using the three conditions of continuity: f(c) is defined, the limit as x approaches c exists, and the limit as x approaches c of f(x) equals f(c).

The value of f(3) is calculated and found to be 5, satisfying the first condition of continuity.

The limits as x approaches 3 from the left and right are calculated and found to be 5 and 1, respectively, indicating that the limit as x approaches 3 does not exist.

Therefore, the function is not continuous at x equals three.