1. 1.6 Continuity Objectives:
To determine continuity of functions
To use 3-step definition in proving continuity of functions

2. What does it mean – “Continuous function”?
A function without breaks or jumps
A function whose graph can be drawn without lifting the pencil 

3. A function can be discontinuous at a point
To be continuous on an interval, a function must be continuous at its Every Point
A function can be discontinuous at a point
A hole in the function and the function not defined at that point
A hole in the function, but the function is defined at that point

4. Continuity at a Point A function can be discontinuous at a point
The function jumps to a different value at a point
The function goes to infinity at one or both sides of the point

5. Definition of Continuity at a Point
A function is continuous at a point x = c if the following three conditions are met
x = c

6. Some Discontinuities are “Removable”!
A discontinuity at c is called removable if …
the function can be made continuous by
defining the function at x = c
or … redefining the function at x = c

7. “Removable” example The open circle can be filled in to make it
Defining the function at x = 1,
y = 2
The open circle
can be filled in
to make it
continuous

8. Non-removable discontinuity
Ex. -1 1

9. Determine whether the following functions are
continuous on the given interval.
yes, it is
continuous
( ) 1

10. removable discontinuity since filling in (1,2)
( )
discontinuous at x = 1
removable discontinuity since filling in (1,2)
would make it continuous.
Define:

11. Which of these are (Dis)Continuous
when x = 1 ?… Why yes or not?
Are any removable?

12. g(x) is continuous at x = 2
Discuss / show continuity of g(x) at x = 2 3 3
g(x) is continuous at x = 2

13. Continuity Theorem
A function will be continuous at any number x = c for which is defined, when
is a polynomial function (at every real number)
is a power function (at every number in its domain)
is a rational function (at every number in its domain)
is a trigonometric function (in domain)

14. Properties of Continuous Functions
If f and g are functions, continuous at x = c, then …
is continuous (where b is a constant)
is continuous

15. One Sided Continuity
A function is continuous from the right at a point x = a if and only if
A function is continuous from the left at a point x = b if and only if a b

16. Continuity on an Interval (Summary)
The function f is said to be continuous on an open interval (a, b) if
It is continuous at each number/point
of the interval
It is said to be continuous on a closed interval [a, b] if
It is continuous at each number/point of the interval, and it is continuous from the right at a and continuous from the left at b

17. Continuity on an Interval (Examples)
On what intervals are the following functions continuous?