Understanding Continuity in Terms of Limits

 

Shelby Smith

 

 

A function is continuous if it is unbroken fro all values of x. The formal definition states that f(x) is continuous if the limit as f(x) approaches a is equal to that of f(a).

 

It is also important to remember that the lim of f(a) exists iff f(x) approaching a from the left is equal to the lim f(x) as x approaches a from the right.

 

 

 

This can also be defined:

 

 

 

Therefore there are three possible problems which cause discontinuity

 

I. f(a) is not defined

II. the limit may not exist

 

 

III. the limit may exist but is not equal to f(a)

 

 

Sample Questions

 

1.

 

 

Image edited from http://www.richland.edu/james/lecture/m116/functions/piecewise1.gif

 

 

What is the

 

What is the

 

 

What is the

 

 

Is the function continuous at f(-1)?

 

2. Suppose that f(x) is continuous at x=3 and that , which statements about f(x) must be true?

 

I. f(3)=17

 

II. 3 is in the domain of f(x)

 

III.

 

 

 

 

 

 3. Is      continuous at x=2?

 

 

 

 

 

 

 

4. Suppose that the function f(x) is continuous at x=2 and that f(x) is defined

 

 

 

What is the value of a?

 

a)2

b)3/4

c)1/2

d)5/8

e)2/3

 

 5. For what values of x is  discontinuous?

 

a)  x=0

b)  x=1

c)  x=-1

d)  x=1 and x=-1

e)  all real numbers

 

 

6.   is continuous for all real numbers EXCEPT:

 

a)  x=0

b)  x=1

c)  x=1 and x=-1

d)  x=-1

e)  x=2

 

 

Answers

1.) 2, 2, does not exist, no

2.) All three

3.) No, althought the limit exists, the function is not continuous

4.) c

5.) b

6.)a

 

 

 

 

For more info:

 

http://www.sparknotes.com/math/precalc/continuityandlimits/section3.rhtml 

 

Sources:

Questions 3 and 4 from calculus text book

Questions 5 and 6 from Amsco's AP Calculus AB/BC by Maxine Lifshitz. NY.2004.