1. Continuity • What is continuity? • Algebraic properties • Compositions

2. Definition • A function f is continuous at x=c if • f(c) is defined • limxc f(x) exits • limxc f(x) = f(c) • If one of these fails to hold, say that f is discontinuous at c.

3. Examples • The absolute value function is continuous everywhere • Polynomials are continuous • Which of the following are continuous at x=3?

4. More Examples

5. Bracket Function

6. Right and Left Continuity

7. Another example

8. Question

9. Non-removable discontinuities

10. Combinations of Continuous Functions

11. Important facts

12. Examples • For which x are the following functions continuous?

13. Compositions • Suppose that limxc g(x)=L and that f is continuous at L. Then limxc f(g(x)) = f(L), i.e. limxc f(g(x)) = f(limxc g(x)) • Consequence: If g is continuous at x=c and if f is continuous at g(c) then f(g(x)) continuous at c (composition of continuous functions is continuous)

14. Problems