Continuity – Part 2

1. Continuity – Part 2

2. The THREE requirements for a function to be continuous at x=c … • C must be in the domain of the function - you can find f(c ), • The right-hand limit must equal the left-hand limit which means that there is a LIMIT at x=c, and • AND

3. Properties of Continuity If b is a real number and f and g are continuous at , then the following functions are already continuous at c… 1. Recall: So if then i.e., is continuous

4. Properties of Continuity If b is a real number and f and g are continuous at , then the following functions are already continuous at c… 2. Recall: So if then i.e., is continuous

5. Properties of Continuity If b is a real number and f and g are continuous at , then the following functions are also continuous at c… 3. Recall: So if then i.e., is continuous

6. Properties of Continuity If b is a real number and f and g are continuous at , then the following functions are also continuous at c… 4. Recall: So if then i.e., is continuous

7. Properties of Continuity If g is continuous at c and f is continuous at g(c), then the composite function ( f◦ g)(x) is also continuous at c. Recall: So if then i.e., ( f◦ g)(x) is continuous

8. Determine whether the function is continuous at (a) x = -1 (b) x = 2.

9. Is the function continuous at -1? f (-1) = 2(-1) + 3 = 1 (REQ#1) (REQ#2) (REQ#3) f (x) is CONTINUOUS at x = -1.

10. Is f (x) continuous at x = 2? f (2) = -2 + 5 = 3 (REQ#3) (REQ#2) There is NO LIMIT (no need to test – does not meet requirement #2!) (REQ#3) f (x) is NOT CONTINUOUS at x = 2.

11. What does the Graph look like? • Quickly graph this piecewise function and see if it confirms our conclusions!

12. This is called a JUMP discontinuity!

13. Polynomial Functions Every polynomial function is continuous at every real number.

14. Functions that are continuous in their respective domains… Rational Functions {x: q(x) ≠ 0} Power Functions if n is odd: all real #s if n is even: {x: x > 0}

15. Functions that are continuous in their respective domains… Logarithmic Functions {x: x > 0} Exponential Functions All Real Numbers

16. Functions that are continuous in their respective domains… Sin(x) or Cos(x) All real numbers Tan(x) or Sec(x) Cot(x) or Csc(x)

17. Functions that are continuous in their respective domains… Arccos(x) or Arcsin(x) -1 ≤ x ≤ 1 Arctan(x) or arccot(x) All real numbers Arcsec(x) or Arccsc (x) |x| ≥ 1

18. Functions and Limits to KNOW!