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. Suppose that the function f(x) is continuous at x=2 and that f(x) is defined
What is the value of a?
4. Is continuous at x=2?
Answers
1.) 2, 2, does not exist, no
2.) All three
3.) 1/2
4.) No, althought the limit exists, the function is not continuous
For more info:
http://www.sparknotes.com/math/precalc/continuityandlimits/section3.rhtml
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