1. Continuity
2.4

2. This function has discontinuities at x=1 and x=2.
Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same as the value of the function. 1 2 3 4
This function has discontinuities at x=1 and x=2.
It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

8. Show g(x)=x^2 + 1 is continuous at x = 1

12. Types of Discontinuities
There are 4 types of discontinuities
Jump
Point
Essential
Removable
The first three are considered non removable

13. Jump Discontinuity
Occurs when the curve breaks at a particular point and starts somewhere else
Right hand limit does not equal left hand limit

14. Point Discontinuity
Occurs when the curve has a “hole” because the function has a value that is off the curve at that point.
Limit of f as x approaches x does not equal f(x)

15. Essential Discontinuity
Occurs when curve has a vertical asymptote
Limit dne due to asymptote

16. Removable Discontinuity
Occurs when you have a rational expression with common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is removable.

17. Places to test for continuity
Rational Expression
Values that make denominator = 0
Piecewise Functions
Changes in interval
Absolute Value Functions
Use piecewise definition and test changes in interval
Step Functions
Test jumps from 1 step to next.

18. Continuous Functions in their domains
Polynomials
Rational f(x)/g(x) if g(x) ≠0
Radical
trig functions

19. Find and identify and points of discontinuity
Non removable – jump discontinuity

20. Find and identify and points of discontinuity
Non removable – essential discontinuity
VA at x = 4

21. Find and identify and points of discontinuity
2 points of disc. (where denominator = 0)
Removable disc. At x = 5
Non removable essential at x = -1 (VA at x = -1)

22. Find and identify and points of discontinuity
Non removable point discontinuity

23. Find and identify and points of discontinuity
2 points of disc. (where denominator = 0)
Removable disc. At x = 5
Non removable essential at x = -4 (VA at x = -4)