1. A function f is continuous at the point x = a if the following are true:
f(a) a

2. Example Continuous everywhere except at
At which value(s) of x is the given function discontinuous?
Continuous everywhere
Continuous everywhere except at

3. F is continuous everywhere else h is continuous everywhere else
and
and
Thus F is not cont. at
Thus h is not cont. at x=1.
F is continuous everywhere else
h is continuous everywhere else

4. Continuous Functions If f and g are continuous at x = a, then
A polynomial function y = P(x) is continuous at every point x.
A rational function is continuous at every point x in its domain.

5. Overview of Problems 1 2 3 4 5

6. Continuity
Problem 1
Solution

7. Continuity
Problem 2
Solution

8. Continuity
Problem 3
Solution

9. Continuity
Problem 4
Answer
Removable
Not removable

10. Continuity Problem 5 Solution
By the intermediate Value Theorem, a continuous function takes any value between any two of its values. I.e. it suffices to show that the function f changes its sign infinitely often.