1. CONTINUITY AND ONE-SIDED LIMITS
Section 1.4

2. When you are done with your homework, you should be able to…
Determine continuity at a point and continuity on an open interval
Determine one-sided limits and continuity on a closed interval
Use properties of continuity
Understand and use the Intermediate Value Theorem

3. CONTINUITY AT A POINT
A function is continuous at c if the following three conditions are met.
is defined.
exists.

4. CONTINUITY ON AN OPEN INTERVAL
A function is continuous on an open interval if it is continuous at each point in the interval. A function that is continuous on the entire real line is everywhere continuous.

5. Is the following function continuous at 3?
Yes No

6. Is the following function continuous at 2?
Yes
No since the limit as x approaches 2 does not exist.
No since the function evaluated at 2 is not equal to the limit as x approaches 2.

7. ONE-SIDED LIMITS means the limit as x approaches c from the right.
means the limit as x approaches c from the left.
One-sided limits are useful in taking limits of functions involving radicals.
If n is an even integer,

8. THE GREATEST INTEGER FUNCTION
The greatest integer function is denoted

9. Evaluate -1 -2
Does not exist

10. THE EXISTENCE OF A LIMIT
Let f be a function and let c and L be real
numbers. The limit of as x approaches
c is L if and only if and

11. Evaluate -1
0.5
Does not exist

12. CONTINUITY ON A CLOSED INTERVAL
A function is continuous on a closed interval if it is continuous on the open interval and and The function f is continuous from the right at a and continuous from the left at b.

13. 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.
Scalar multiple
Sum and difference
Product
quotient

14. Polynomial Rational Radical Trigonometric
EXAMPLES OF FUNCTIONS WHICH ARE CONTINUOUS AT EVERY POINT IN THEIR DOMAINS
Polynomial
Rational
Radical
Trigonometric

15. CONTINUITY OF A COMPOSITE FUNCTION
If g is continuous at c and f is continuous at , then the composite function given by is continuous at c.

16. THE INTERMEDIATE VALUE THEOREM
If is continuous on the closed interval and d is any number between and , then there is at least one number c in such that

17. A VISUAL OF THE IVT

18. CONSIDER THE FUNCTION BELOW ON THE GIVEN INTERVAL
CONSIDER THE FUNCTION BELOW ON THE GIVEN INTERVAL. IF THE IVT CAN BE APPLIED USE IT TO FIND THE INPUT VALUE WHICH YIELDS AN OUTPUT OF ½.
The IVT cannot be applied since the function is not continuous.
The IVT can be applied since the function is continuous on the given interval.