Understanding Continuity in Terms of Limits

Shelby Smith

A function is continuous if it is unbroken fro all values of x. T he 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 tree 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. Suppose that the function f(x) is continuous at x=2 and that f(x) is defined

What is the value of a?

Answers

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

2.) All three         

3.) 1/2

For more info:

http://64.233.169.104/search?q=cache:4gvwCaS-djIJ:community.middlebury.edu/~abbott/UA/UA-4-1.pdf+continuity+in+terms+of+limits&hl=en&ct=clnk&cd=5&gl=us

http://www.sasked.gov.sk.ca/docs/calc30/unit_c.htm

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